Arithmetic Practice Research Articles

Affective and cognitive factors associated with Chinese and Italian children’s arithmetic performance

BackgroundThis study aimed to investigate the cognitive and affective factors associated with cross-cultural differences in arithmetic tasks.MethodsA total of 404 third- and fourth- graders were recruited from China and Italy to complete exact arithmetic, arithmetic estimation and cognitive tasks (i.e., short-term memory, executive functions, and fluid reasoning). Their mathematical anxiety was also measured.ResultsThe results showed that Chinese children performed better than Italian children in both arithmetic tasks and in shifting task. Italian children performed better in visuospatial updating task and reported higher levels of mathematical anxiety than their Chinese peers. Multi-group path analyses showed that the patterns of relations among cognitive factors (i.e., short-term memory, inhibition and shifting), mathematical anxiety, and arithmetic performance were similar across groups. The only exception was that visuospatial updating uniquely predicted arithmetic estimation for Chinese but not for Italian children.ConclusionsChinese children outperformed their Italian peers in the exact arithmetic task, likely due to the greater emphasis on arithmetic fluency in Chinese mathematics education, both in schools and at home. They also had a slight advantage than Italian peers in the arithmetic estimation task. The unique link between updating and arithmetic estimation found in Chinese children but not Italian children suggests that, although arithmetic estimation is not emphasized in the curricula of either country, instruction and practice in exact arithmetic may enhance Chinese children’s efficiency in solving arithmetic estimation problems.

Explaining procedures and interleaving practice in fraction arithmetic

BackgroundExplicit instruction is a pedagogical approach in which students are taught step-by-step procedures for solving problems and practice using the procedures. Despite considerable evidence supporting the use of this general approach, relatively little is known about how varying specific design features impact its effectiveness. AimsThis preregistered study investigated the impact of two design features, explaining rationales for procedures and practicing in interleaved sequence, on the efficacy of explicit instruction in fraction arithmetic. SampleParticipants were 156 sixth to ninth grade children. MethodsChildren received a brief intervention involving explicit instruction in fraction arithmetic procedures. They were randomly assigned to receive or not receive explanations of rationales for the procedures, and to practice different procedures in either interleaved or blocked sequence. ResultsRelative to pretest, accuracy improved on immediate and delayed posttests (Cohen's d = 0.88 and 0.55 respectively). The magnitude of this improvement was not affected by either experimentally-manipulated factor or by both of them together. ConclusionsThe results support the use of explicit instruction in fraction arithmetic, but provide no evidence that explaining rationales for procedures, or interleaving practice thereof, increases the effects of such instruction on procedural knowledge.

Improving Children’s Understanding of Mathematical Equivalence: An Efficacy Study

A vast majority of elementary students struggle with the core, pre-algebraic concept of mathematical equivalence. The Improving Children’s Understanding of Equivalence (ICUE) intervention integrates four research-based strategies to improve outcomes for second grade students: (1) introducing the equal sign before arithmetic, (2) nontraditional arithmetic practice, (3) concreteness fading exercises, and (4) comparison and explanation. In a large-scale randomized control trial in California public schools, 132 second grade teachers were randomly assigned to either use the ICUE intervention or an active control consisting of nontraditional arithmetic practice alone. Using data from 121 teachers in the analytic sample, the study found that students in the intervention group outperformed students in the active control on proximal and transfer measures of equivalence with no observable tradeoffs in computational fluency. The findings suggest that the ICUE intervention helps students construct a robust understanding of mathematical equivalence, a critical precursor to success in algebra.

A conventional and digital mathematical board game design and development for use by students in learning arithmetic

This paper reported the design and development of a conventional and digital mathematical board game for use by students in learning arithmetic. At the time of research, there is no significant indication that a mathematical board game is available in scientific and published patent documentation. The availability of mathematical board games for students’ drills and practice in arithmetic, especially in mathematical statement construction, would benefit them, as this competency is an essential life skill. This research was conducted through the design and development research method with the procedure of users’ need analysis, researcher as developer capability analysis, product design, product development, field testing in its natural setting environment, and the prototype. The board game prototype was developed in both conventional printed and digital versions. The field testing for the conventional printed version was conducted at secondary school classes with 34 and 36 students, respectively, while for the digital by selected participants. The field testing shows that the developed mathematical board game can work as expected in its natural setting environment.

Distributions of textbook problems predict student learning: Data from decimal arithmetic.

This study investigated relations between the distribution of practice problems in textbooks and students’ learning of decimal arithmetic. In Study 1, we analyzed the distributions of decimal arithmetic practice problems that appeared in 3 leading math textbook series in the United States. Similar imbalances in the relative frequencies of decimal arithmetic problems were present across the 3 series: Addition and subtraction more often involved 2 decimals than a whole number and a decimal, but the opposite was true for multiplication and division. We expected children’s learning of decimal arithmetic to reflect these distributional biases. In Studies 2, 3, and 4, we tested the prediction that children would have more difficulty solving types of problems that appeared less frequently in textbooks, regardless of the intrinsic complexity of solving the problems. We analyzed students’ performance on decimal arithmetic from an experiment conducted in a different lab 35 years ago (Study 2), from a contemporary large-scale web-based learning platform (Study 3), and from a recent controlled experiment conducted in our own lab (Study 4). Despite many differences among the 3 studies, performance in all 3 was in accord with the predictions. These findings suggest that the distributions of practice problems in math textbooks may influence what children do and do not learn. Usefulness of analyzing textbook problem distributions as well as educational implications of the current findings are discussed. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

A polyomino puzzle for arithmetic practice

Recent trends in mathematics education emphasize discovery learning over drill. This has proven to be a bad idea in some cases, for the simple reason that practice is required to learn basic arithmetic skills. Drills in arithmetic skills can be made interesting through gamification. This study proposes a family of puzzles that gamify arithmetic practice. The puzzles are designed with an evolutionary algorithm forming an instance of automatic content generation. Two methods of evolutionary puzzle design are presented and discussed. The first method used transformed the problem into an almost trivial optimization. The second algorithm was designed to avoid the flaws of the first and produced a huge variety of puzzles. A hardness measure, based on the difficulty experienced by the evolutionary puzzle generator, is employed. The hardness measure is tested on a large collection of puzzles produced with the evolutionary automatic content generation system. An initial assumption, that all the pieces in the puzzle must be used to achieve a maximum score, was shown to be incorrect in puzzles located via automatic search. Two classes of puzzle are defined: those where the optimal solution uses all pieces and those where the optimal solution fails to use at least one piece. The latter sort of puzzle were found to be far more common in the search space.

Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts

Numerous recent discoveries in China of ancient tombs have greatly increased our knowledge of ritual and religious practices. These discoveries include excavated oracle bones, bronze, jade, stone and pottery objects, and bamboo manuscripts dating from the twelfth to fourth century BCE. Inscribed upon these artifacts are a large number of records of numerical sequences, for which no explanation has been found of how they were produced. Structural links to the Book of Changes, a divination manual that entered the Confucian canon, are evident; yet, the algorithm described therein dates to the slightly later second to first century BCE. By combining archeological and statistical evidence, we propose a new methodology that enables us to reconstruct and test cleromantic techniques which can explain how these numerical sequences were generated. Dice and divination stalk use, either in combination or separately, appear in fact to have been underlying the rather stable numerical patterns in ancient China all the way back to the late Shang dynasty (1300–1046 BCE). Bringing to light such a long-standing technique, which awaits further confirmation from the ever-growing database of newly discovered numerical and textual records, can change drastically our understanding of early Chinese history and of the historical development of sophisticated arithmetical practices and the rationalization of chance.

Improving children’s understanding of mathematical equivalence via an intervention that goes beyond nontraditional arithmetic practice.

Elementary school children (ages 7–11) struggle to understand mathematical equivalence, a foundational prealgebraic concept. Some manipulations to the learning environment, including well-structured nontraditional arithmetic practice alone, have been shown to improve children’s understanding; however, improvements have been modest. The goal of this study was to test an iteratively developed supplemental intervention for second grade that was designed to yield widespread mastery of mathematical equivalence. The intervention included three components beyond nontraditional arithmetic practice: (a) lessons that introduce the equal sign outside of arithmetic contexts, (b) “concreteness fading” exercises, and (c) activities that require children to compare and explain different problem formats and problem-solving strategies. After the development process, a small, randomized experiment was conducted with 142 students across eight second grade classrooms to evaluate the effects of the intervention on children’s solving of mathematical equivalence problems, encoding of mathematical equivalence problems, and defining of the equal sign. Classrooms were randomly assigned to the intervention or to nontraditional arithmetic practice alone. Analyses at the classroom level demonstrated that the intervention classrooms performed better than the active control classrooms both in terms of pre-to-post change in understanding of mathematical equivalence (g = 1.87) and accuracy on transfer problems at posttest (g = 2.07). Nonparametric analyses led to the same conclusions. Results suggest that the comprehensive intervention improves children’s understanding of mathematical equivalence to levels that surpass equivalent levels of well-structured arithmetic practice alone, as well as business-as-usual benchmarks from previous studies. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

The cerebral, extra-cerebral bodily, and socio-cultural dimensions of enculturated arithmetical cognition

Arithmetical cognition is the result of enculturation. On a personal level of analysis, enculturation is a process of structured cultural learning that leads to the acquisition of evolutionarily recent, socio-culturally shaped arithmetical practices. On a sub-personal level, enculturation is realized by learning driven plasticity and learning driven bodily adaptability, which leads to the emergence of new neural circuitry and bodily action patterns. While learning driven plasticity in the case of arithmetical practices is not consistent with modularist theories of mental architecture, it can be enriched by the theory of neural reuse. According to neural reuse, cerebral regions are reused to contribute to multiple neural circuits in functionally constrained ways throughout ontogeny. By hypothesis, learning driven plasticity is complemented by learning driven bodily adaptability, which suggests that there is an interesting functional relationship between finger gnosis, finger counting, and arithmetical practices. The emerging perspective on enculturated arithmetical cognition will be complemented by considerations on associated developmental and acquired disorders, namely developmental dyscalculia and acquired acalculia. The upshot is that we need to take the cerebral, extra-cerebral bodily, and socio-cultural dimensions of enculturation into account in order to arrive at a better understanding of the phylogenetic and ontogenetic conditions of arithmetical cognition.

Arithmetic practice that includes relational words promotes understanding of symbolic equations

We tested whether using relational words to highlight the relational nature of equality during arithmetic practice can improve what children learn from that practice. Children were randomly assigned to one of four addition practice conditions: (a) relational words: equality symbols were sometimes replaced with relational words [e.g., “is the same amount as”], (b) traditional symbols only: equality was only symbolized with the equal sign or equal bar, (c) operational words: equality symbols were sometimes replaced with operational words [e.g., “adds up to”], or (d) no extra practice. As hypothesized, the relational words group showed evidence of a more relational understanding of symbolic equations at posttest: they were more likely than the other groups to encode the “=” when asked to reconstruct pre-algebra problems, to show transfer from addition practice to subtraction, and to use advanced solving strategies. Results suggest children can benefit from minor arithmetic practice modifications that highlight the relational aspects of equality.

Protocol for a systematic review: Interventions to improve mathematics achievement in primary school-aged children: a systematic review.

Protocol for a systematic review: Interventions to improve mathematics achievement in primary school-aged children: a systematic review.

“Set Them to the Cyphering Schoole”: Reading, Writing, and Arithmetical Education, circa 1540–1700

AbstractDuring the late sixteenth and seventeenth centuries, English men and women replaced their existing oral and object-based arithmetical practices with literate practices based on Arabic numerals. While the adoption of Arabic numerals was incentivized by continental commercial developments, this article argues that England's increasing literacy rates and the development of vernacular arithmetic textbooks enabled changing arithmetical practices. By exploring the qualities of printed books, analyzing marginalia in arithmetic textbooks, and examining changing educational advertisements and curricula over time, we can demonstrate the importance of literacy and literature to early modern arithmetical education.

「数と計算」領域を通じて算数科の深化を目指した指導実践 : 小数のかけ算とわり算の指導を中心に(高田由夫先生古希記念号)

「数と計算」領域を通じて算数科の深化を目指した指導実践 : 小数のかけ算とわり算の指導を中心に(高田由夫先生古希記念号)

Incalculable, Unaccountable, Indivisible? Walt Whitman's Lessons in Arithmetic

I heartily advise you to peg away at the arithmetic-do something at it every day-arithmetic is the foundation of all such things-(just as a good stone wall is the foundation for a house)- become a good arithmetician first of all--Whitman to Pete Doyle1As surprising as it might seem to some, Walt Whitman was once a math teacher. The proto-hippie with his flowing beard and saggy pants, so proudly singing of infinites and things beyond measure, at one point stood in front of a crowded roomful of students in a stuffy Long Island schoolhouse and discussed multiplication tables, long division and subtraction rules. And he was passionate about it, too2-a passion he carried forward into Leaves of Grass and inscribed into virtually every page of his poetic corpus.Over the course of Whitman's poetic life, arithmetic in Leaves did not remain a static symbol but changed alongside the poet's project. Embraced for its radical language of equality, the science of addition, subtraction, multiplication, and division quickly became a metaphoric staple in Whitman's poetic assembling and disassembling of nature into temporary aggregates and complex particulars. Still, with the dawning of the Civil War, Whitman observed the mathematical principles he employed as a lyrical elevation of life, turn into a practical devaluation of it-a turning into numbers of bodies, that could then easily be calculated away as long as the overall equation was favorable. What Whitman the educator embraced and Whitman the celebrated, the Good Grey Poet schooled by fratricidal conflict had to reign back in. The following pages will follow this development from early traces of mathematical thought in Whitman to its inception as a poetic-political program in the 1855 edition of Leaves of Grass and its subsequent reevaluation in the context of the Civil War.While grounded in a historical reading of Whitman's text, this essay then also echoes and corresponds to an emerging critical discourse on the intersections of art and mathematics3-especially in German media theory following the publication of Friedrich Kittler's post-structuralist magnum opus Musik und Mathematik, which media-archeologically tries to rethink the epistemological walls that today separate letter, sound, and number. While Kittler himself dives deep into a Homeric past to stake his claims, the nineteenth century-and especially Whitman's democratic epic-seem like a prime entry point to considering these seemingly divergent scribal and logical systems together. Growing out of a soil saturated with New England's transcendental idealism, the mid-1800s saw number and letter suddenly rediscovering their kinship. Calculations are works of taste in the same manner as the productions in music and for instance a 1840s article in a musical journal states, though it may be also true that they are more often applied than music or poetry to what are commonly called purposes.4 Whitman, all about useful poetry, would take up the challenge suggested by such a claim and (re)construct a rigorous arithmetic in and out of language that even by today's standard has lost little of its radicalism. And it all began with Whitman's own, hands-on experience with mathematics: in school.Arithmetic of YouthWhile Walter Whitman's time as a small-town schoolteacher (18361841) was by any stretch of the imagination just a temporary solution to help the ambitious young man through a financial rough patch, he did apparently show an honest interest in the well-being of his pupils as well as employ some rather unorthodox teaching methods-one of them being the daily practice of mental arithmetic. In those days, as Bernstein Freedman observes, this practice was evidently a procedure strange enough to warrant special comment when Whitman's former students were interviewed by Whitman' disciples (in this case Horace Traubel) many years later (Freedman 29).In this regard, Whitman seems at least partially indebted to the set of educational practices he had experienced as a student himself. …

Arithmetic practice can be modified to promote understanding of mathematical equivalence.

This experiment tested if a modified version of arithmetic practice facilitates understanding of math equivalence. Children within 2nd-grade classrooms (N 166) were randomly assigned to practice single-digit addition facts using 1 of 2 workbooks. In the control workbook, problems were presented in the traditional “operations answer” format (e.g., 4 3 __) and were organized pseudorandomly throughout the workbook. In the modified workbook, problems were presented with operations on the right side (e.g., __ 4 3), the equal sign was sometimes replaced by relational words (e.g., “is equal to”), and problems were organized by equivalent sums such that several problems in a row would all have the same sum. Children who used the modified workbook constructed a better understanding of math equivalence than did children who used the control workbook. This advantage persisted approximately 5–6 months after the practice had ended, and there were no observed “trade-offs” with computational fluency. A mediation analysis showed that the modified practice improved understanding as intended by decreasing children’s reliance on operational patterns. Results suggest that small differences in the organization and format of arithmetic practice can yield substantial differences in children’s understanding of fundamental mathematics concepts.

Arithmetic, Mathematical Intuition, and Evidence

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.

Activation of Operational Thinking During Arithmetic Practice Hinders Learning And Transfer

Many children in the U.S. initially come to understand the equal sign operationally, as a symbol meaning “add up the numbers” rather than relationally, as an indication that the two sides of an equation share a common value. According to a change-resistance account (McNeil & Alibali, 2005), children's operational ways of thinking are never erased, and when activated, can interfere with mathematics learning and performance, even in educated adults. To test this theory, undergraduates practiced unfamiliar multiplication facts (e.g., 17-times table) in one of three conditions that differed in terms of how the equal sign was represented in the problems. In the operational words condition, the equal sign was replaced by operational words (e.g., multiplies to). In the relational words condition, the equal sign was replaced by relational words (e.g., equivalent to). In the control condition, the equal sign was used in all problems. The hypothesis was that undergraduates' fluency with practiced facts and transfer problems would be hindered in the operational words condition compared to the other conditions. Results supported this hypothesis, indicating that the activation of operational thinking is indeed detrimental to learning and transfer, even in educated adults.

Comparative ERP and behavior study of ADHD and normal children in mental arithmetic practice

Objective To compare the differences of event-related potentials(ERP) of left and right cerebral hemispheres and behavioral characteristics in the mental arithmetic practice between ADHD and normal children, and to explore the neural mechanisms of mental arithmetic by ADHD children. Methods ERP and behavioral data of 32 ADHD and 32 normal children in mental arithmetic, to illustrate, simple addition, subtraction and multiplication within 20 were recorded. Results ERP: ① The differences of N2 amplitudes in ADHD children 's left and right cerebral hemispheres while doing mental arithmetic, such as, addition, subtraction and multiplication were not statistically significant (addition: (-6.2±7.3)μV, (-6.6±8.1)μV; subtraction: (-5.5±6.4)μV, (-5.8±6.5)μV; multiplication: (-5.9±8.2)μV, (-6.0±8.6)μV; all P>0.05). However, normal group's N2 amplitude of left hemisphere(F3) were obviously higher than those of the right hemisphere(F4), considered to be statistically significant difference (addition: (-6.8±4.0)μV, (-5.9±4.3)μV; subtraction: (-7.5±4. 8)μV, (-6.3±4.4)μV; multiplication: (-6.9±3.7)μV, (-5.7±4.4)μV; all P 0.05). Behaviorism: ① ADHD group's reaction time towards answer making were much longer than those of the normal control group (addition: (983±183)ms, (833±164)ms; subtraction: (1 005±160)ms, (859±170)ms; multiplication: (975±180)ms, (836±175)ms; P<0.05). ② Accurate rate : ADHD group were all lower than those of normal group. Conclusion There were functional limitations of no obviously advantage hemisphere and two hemisphere coordination while ADHD children were performing mental arithmetic, also limitations in conflicting information processing, decision making, and more obviously in subtraction practice. Key words: ADHD children; Mental arithmetic; Left and right Cerebral hemispheres; Event-related potentials; Behaviorism

Lagrange and the four-square theorem

In this article we will first provide an overview of Lagrange’s arithmetic work, written between 1768 and 1777. We then focus on the proof of the four-square theorem published in 1772, and shed light on the context of its implementation by relying on other memoirs by Lagrange and Leonhard Euler. By means of this analysis, our goal is to give a concrete vision of the arithmetic, algebraic and analytical methods and tools used by Lagrange in number theory and place his arithmetic practice in the context of the second half of the eighteenth century.

It pays to be organized: Organizing arithmetic practice around equivalent values facilitates understanding of math equivalence.

This experiment tested the hypothesis that organizing arithmetic fact practice by equivalent values facilitates children’s understanding of math equivalence. Children (M age 8 years 6 months, N 104) were randomly assigned to 1 of 3 practice conditions: (a) equivalent values, in which problems were grouped by equivalent sums (e.g., 3 4 7, 2 5 7, etc.), (b) iterative, in which problems were grouped iteratively by shared addend (e.g., 3 1 4, 3 2 5, etc.), or (c) no extra practice, in which children did not receive any practice over and above what they ordinarily receive at school and home. Children then completed measures to assess their understanding of math equivalence. Children who practiced facts organized by equivalent values demonstrated a better understanding of math equivalence than children in the other 2 conditions. Results suggest that organizing arithmetic facts into conceptually related groupings may help children improve their understanding of math equivalence.