Edited by: Douglas F. Kauffman, Independent Researcher Boston, United States
Reviewed by: Firat Soylu, University of Alabama, United States; Michael S. Dempsey, Boston University, United States
This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology
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Abacus mental arithmetic involves the skilled acquisition of a set of gestures representing mathematical algorithms to properly manipulate an imaginary abacus. The present study examined how the beneficial effect of abacus cothought gestures varied at different skill and problem difficulty levels. We compared the mental arithmetic performance of 6 to 8yearold beginning (
Abacus arithmetic is an ideal model for examining the changing beneficial effect of cothought gestures for learning mathematics at different skill and problem difficulty levels. According to Gesture as Simulated Action theory, learners spontaneously gesture to activate motor programs that assist working memory as imagery tasks pass a threshold of difficulty (
The role of cothought gestures, i.e., noncommunicative hand movements without accompanying speech, in mathematics learning is less well understood than cospeech gestures. In two recent papers, cothought gestures were found to change from action simulation to representation of action plans (
No studies have examined the possible beneficial effect of cothought gestures in learning mathematics. Early studies of abacus mental arithmetic have all noted that learners move their hands and fingers when performing mental calculations, as if manipulating a real abacus (
The abacus is an historically significant cultural artifact that affects how the brain processes calculations as part of a living tradition of embodied mathematics. Abacus arithmetic has become one of the most common and widespread forms of early childhood mathematics education throughout Asia. In the highly competitive educational systems of countries like Singapore, Taiwan, and South Korea, nearly all children attend supplemental classes till late in the evening. The trend has been for children to start at an increasingly early age, with most beginning around 5 or 6 years old.
Abacus mental arithmetic is taught multimodally, as both visual and motor operations. The abacus uses a finite set of rules or algorithms for moving the abacus beads to perform addition or subtraction of single digits (Supplemental Materials 3: How to Use an Abacus and Example Problems with Gesture Solutions). As
Although mental arithmetic is practiced at all stages, learners are allowed to use different physical aids during training. In a beginning abacus class, children primarily learn by manipulating a physical abacus using correct hand movements. An abacus is an array of beads with five beads in each column. There is a single upper row of beads separated by a horizontal bar from four lower rows of beads. When a bead in the upper row is pushed downward with the index finger to touch the horizontal bar, it registers a digit value of five for that column. When one of the beads in the four lower rows is pushed upward with the thumb to touch the horizontal bar, it registers a digit value of one for that column. Hence, each column can register a digit value from 0 to 9. When a column is designated as the one’s column (×10^{0}), each successive column to the left is a successive power of ×10^{n} and each column to the right a successive negative power of ×10^{n}. Any arithmetic problem can be solved by concatenating a fixed sequence of these hand movements to move the beads according to algorithms for complements of 5 and 10 (see
As a transitional or intermediate stage of instruction, children use a picture card or static diagram of an abacus. Instead of moving beads on a physical abacus, learners touch the picture card or spontaneously gesture over it. This reduces the physical tool to an abstract mathematical diagram or sign. At the advanced stage, learners solve problems purely mentally without a physical abacus or any visual aid. However, as problem size and complexity of operations increase, learners may revert back to using one of the physical aids.
Learners at all stages of instruction, especially at intermediate and advanced performance levels, spontaneously produce cothought gestures which closely mimic the hand movements when using a physical abacus. These movements can sometimes be exaggerated or vary in form. It is important to note that teachers instruct learners on the proper hand movements when using a physical abacus; but they do not instruct learners on how to spontaneously gesture. The only criterion is to solve the problems mentally. Thus, it is totally the learners’ decision to gesture or not. Overall, as abacus learners acquire mental arithmetic skill, they rely less on a physical abacus to perform mental arithmetic. They gradually internalize or embody the abacus tool. Whether and how these spontaneous cothought gestures facilitate mental arithmetic remains unclear.
The present study examined whether the beneficial effect and form of abacus cothought gestures are different among beginning, intermediate, and advanced learners who are asked to solve one, two, and threedigit arithmetic problems. We tested the abacus mental arithmetic performance of 6 to 8yearold beginning, intermediate, and advanced learners. Each learner was randomly assigned to one of three conditions for performing calculations at three difficulty levels (onedigit, twodigit, and threedigit): physical abacus, handsfree (spontaneous gesture) mental arithmetic, and handsrestricted mental arithmetic.
Based on the current state of research, there are several competing predictions about how abacus cothought gestures may benefit learners at different skill levels when solving problems of varying degrees of difficulty. According to Image Maintenance theory, cothought gestures, as bodily acts, should refresh the mental image of an abacus on a “visuospatial scratchpad” (
In contrast, we hypothesize that abacus cothought gestures facilitate mental arithmetic as motor programs that complement visualspatial representation to reduce working memory load. We predict that abacus cothought gestures will be beneficial only for learners who have acquired motor skills that closely reflect simulated action on a physical abacus. In other words, contrary to Image Maintenance theory, we predict that the beneficial effect of spontaneous abacus gestures will vary depending on skill level. And, contrary to the action generation hypothesis, more advanced learners’ spontaneous gestures will more accurately mimic action on a physical abacus.
According to GSA theory, spontaneous gestures activate motor programs that assist working memory as imagery tasks pass beyond a threshold of difficulty. As
Additionally, numerous studies have shown that motorspatial representation complements visualspatial representation by encoding visualspatial sequences as motor plans to reduce working memory load (
We thus expected that abacus learners would master how to use a physical abacus before becoming proficient in using abacus cothought gestures. This is because being able to see the beads on a physical abacus as a visualspatial sequence may demand less working memory compared to maintaining a motorspatial mental representation of it. Beginning learners should be able to perform calculations well with a physical abacus. However, without the aid of a physical abacus, they would perform poorly under the spontaneous gesture condition and handsrestricted condition. Moreover, the accuracy of their gesturing would be poor because they had not yet acquired the motorprograms for abacus gestures.
Intermediate learners would perform mental calculations equally well using spontaneous gestures compared to using a physical abacus for simple one and twodigit problems. Intermediate learners’ gestures should also be highly accurate, following closely the same types of hand movements for moving beads on a physical abacus. In other words, intermediate learners would have acquired motorprograms for abacus gestures to aid in visualspatial representation of the mental abacus. However, as the demands on working memory increase with problem difficulty, the beneficial effect of cothought gestures in reducing working memory load should attenuate for the most difficult problems. Intermediate learners, may thus perform less well using spontaneous gestures compared to physical abacus for more difficult threedigit problems. Moreover, the beneficial effect of cothought gestures should be most salient when comparing intermediate learners’ ability to perform mental arithmetic under the handsfree spontaneous gesture to that in the handsrestricted condition. These learners would perform poorly in the handsrestricted condition because they had not yet fully automated and internalized the motor programs for abacus gestures to maintain the mental representation of an imaginary abacus.
Different from beginning and intermediate learners, advanced learners would have fully internalized and automated abacus gesture motorprograms. Thus, abacus cothought gestures should not only have a beneficial effect for mental arithmetic performance but also show increasing movement accuracy similar to manipulating a physical abacus. Hence, advanced learners should not even need overt gesturing to refresh motorspatial mental representation. They would be able to perform mental arithmetic calculations without much conscious effort through the assistance of gestures. Advanced learners’ gestures should be highly accurate and their calculation scores in the handsfree and handsrestricted conditions should be comparably high and nearly as high as with a physical abacus.
Evidence for these predictions comes from previous abacus studies.
Understanding how the beneficial effect of abacus gestures changes at different levels of skill and problem difficulty can provide us with insights into the role of visual and motor working memory in abacus mental arithmetic. Some studies have shown that advanced abacus learners perform significantly better on mental arithmetic tasks compared to untrained controls (
Neurophysiological studies of abacus mental arithmetic show activation in cortical areas important for both visual and motor imagery. Activation occurs in the parietal cortex (
The current study also sheds light on how gesture may assist in transitioning from concrete objects to mental representation when learning arithmetic. The use of concrete manipulatives has been a staple of early mathematics education for decades (
Theoretical models such as Bruner’s for mathematics learning has been widely applied in curricula, especially for early childhood education (
Despite widespread implementation of the enactiveiconicsymbolic model in school curricula, little work has explored the mechanisms underlying how learners shift from embodied concrete perception and action to abstract concepts. This lack of explanation has further led to controversy over whether concrete manipulatives are even effective. Some studies have shown that in certain circumstances instruction with concrete manipulates led to worse performance (
One hundred and eighty children (half males) participated in this experiment from 2010 to 2011. They were English speaking Singaporeans and attended abacus classes at the Classical Mental Arithmetic School (CMA). CMA was one of the popular schools in Singapore teaching young learners abacus mental arithmetic using two hands and four fingers. It has 21 branches in Singapore. In the present study, we collected data in five of them. On average, children were 7;1 (years;months) years old, ranging from 5;11 to 8;1 years old. All of them were typical primary school students. Abacus training is common among Singaporean children, across socioeconomic and educational backgrounds. The selected participants were thus a representative sample. We further note that children at this young an age are at the very beginning stage of their mathematics education in primary school. We chose a narrow age range to minimize the influence of the students’ regular school education. All the procedures were approved by the institutional review board of the authors’ university at the time of the study, in compliance with the Declaration of Helsinki. We obtained the parents’ informed consent prior to the study. The first author presented preliminary work for this article at the workshop, Culture and Cognition in Asia II: Performative Gesture in Religion and Science (17 June 2010 at the National University of Singapore,
Each child was classified into one of the following three categories: beginning learners (
We then randomly assigned learners from each skill level to one of the following three conditions: (1) physical abacus; (2) handsfree mental arithmetic (spontaneous gesture); (3) handsrestricted mental arithmetic.
Demographic information of participants in all conditions.
Physical Abacus 
HandsFree Mental Arithmetic 
HandsRestricted Mental Arithmetic 


Beginning 
Intermediate 
Advanced 
Beginning 
Intermediate 
Advanced 
Beginning 
Intermediate 
Advanced 

Mean  6;11  6;11  7;2  6;11  7;2  7;1  7;1  7;2  6;10 
Age (year; month)  
Gender (male)  8  12  11  9  15  10  8  13  10 
We refer to mental arithmetic as MA. Learners in all conditions were tested individually at their CMA branch and asked to solve 60 addition and subtraction questions (20 onedigit, 20 twodigit, and 20 threedigit). All questions were designed by teachers in CMA. Learners were given 30 min to complete the test, which was ample time for all to finish the problems that they were able to do. However, learners who found that the problems were too difficult to manage could stop at any time. The entire experiment was videotaped. Each child was closely monitored by an experimenter, oneonone, for compliance. None of the children moved his/her hands in the handsrestricted mental arithmetic condition.
Learners in the physical abacus condition solved the problems using a physical abacus, which was the same as the one they used in their regular class. Learners in the handsfree MA (spontaneous gesture) condition solved the same problems, but without the assistance of an abacus. With prompting, they were able to spontaneously move their hands to perform mental calculations. Learners in the handsrestricted MA condition also solved the same problems using mental calculation, but were restricted from moving their hands by holding a ball with both hands.
We calculated the mean proportions of questions with correct answers, which were calculated as the number of correct answers separately divided by the total number of questions at each digitlevel in each group of learners in each condition.
A teacher at CMA then coded the abacus hand movements and abacus gestures produced by learners in the physical abacus condition and handsfree mental arithmetic condition, respectively. Teachers at CMA were well trained in identifying the abacus hand movements and gestures produced by their students. There are two kinds of hand movements: abacus hand movements, produced when manipulating a physical abacus; and abacus hand gestures, produced while doing mental calculation. The teacher coded both kinds of hand movements using a standard answer key, which provided the sequence of gestures to solve each problem. Supplemental Materials 1: Abacus Gestures 1–70, provides a list of illustrations for each of the 70 twohanded gestures and the arithmetic operations they perform. Supplemental Materials 2: The Abacus Hand Movement Lexicon and Correct and Incorrect Gestures, provides the basic 16 hand movements (including no movement) using the left and right index fingers and thumbs. These movements are combined to form the 70 abacus gestures. Supplemental Materials 2 also has examples of correct and incorrect gestures. Supplemental Materials 3: How to Use an Abacus and Example Problems with Gesture Solutions, explains of how to use an abacus and gives examples of onedigit and twodigit problems with the sequence of gestures to compute the answers. We counted the number of abacus hand movements or abacus hand gestures produced per question.
After identifying a gesture, the teacher determined whether the gesture was correct. We sought to understand how learners at different skill levels employed correct gestures or other movements in mental calculations and how the sequence of these correct gestures compared to hand movements when manipulating a physical abacus. Learners are taught in abacus classes stylized or pedagogically correct hand movements using the index finger and thumb up or down in a single column, either as one hand or as two hands in adjacent columns. When learners perform mental arithmetic, they often spontaneously gesture in the air, mimicking these hand movements to move the beads on physical abacus. Each gesture has specific algorithmic meaning depending on context and is executed in a fixed sequence of gestures to solve a particular mathematical problem. However, learners sometimes do not use these correct hand movements or gestures and make mistakes, such as incorrectly moving their index fingers and thumbs, skipping or combining movements, or using fingers other than the index fingers and thumbs. We compared the proportion of correct gestures produced in the handsfree MA (spontaneous gesture) condition to the proportion of correct hand movements in the physical abacus condition. The proportion of correct hand movements or gestures was calculated as the total number of correct hand movements or gestures divided by the total number of fixed algorithmic steps.
To assess intercoder reliability for the coding of the abacus hand movements/gestures and that of the correct gestures, we randomly selected twelve children (three in each condition) for independent coding by a second trained coder. The coder was also one of the teachers at CMA and she was naive to our hypotheses. The interrater agreement was 0.96 (
We examined whether the facilitating role of gesture in solving arithmetic problems varied with the level of abacus skills and the difficulty of problems. We first examined how learners with different levels of abacus skills gestured, by looking at whether these gestures were correct, i.e., following the form of hand movements on an abacus taught in class. We next examined the proportions of correct answers. We investigated these proportions as functions of the method of calculation, level of abacus skills of learners, and level of problem difficulty. The accuracy rate, as the proportion of questions answered correctly, was calculated as the total number of questions answered correctly divided by the total number of questions.
Abacus hand movements produced in the physical abacus condition and abacus gestures produced in the mental arithmetic condition were classified into two categories: correct and incorrect. The proportion of correct abacus hand movements or abacus gestures was calculated as the total number of correct abacus hand movements divided by the total number of abacus gestures possibly produced.
Mean proportions of correct hand movements in the physical abacus condition and correct gestures in the handsfree mental arithmetic condition for one, two, and threedigit problems at beginning, intermediate and advanced skilllevels.
Given the significant three–way interaction, we separately looked at the differences in the proportions of correct abacus hand movements or abacus gestures produced in the physical abacus and the mental arithmetic conditions among three groups of learners. As for beginning learners, we found a significant effect for the problem difficulty,
We next examined the proportions of onedigit, twodigit, and threedigit questions answered correctly in three groups of learners in three different conditions.
Mean proportions of questions answered correctly in one, two and threedigit questions for beginning, intermediate and advanced learners in three conditions: physical abacus, handsfree mental arithmetic, and handsrestricted mental arithmetic.
We ran a repeated measures ANOVA with the difficulty of problems as the independent withinsubject variable, condition and skill level as the independent betweensubject variables, and the proportion of questions answered correctly as the dependent variable. We found a significant effect for the problem difficulty,
For intermediate learners, there were significant effects for problem difficulty,
The findings in the advanced learners were similar to those in the intermediate learners. There were significant effects for problem difficulty,
Our results showed that the beneficial effect of abacus gestures on the accuracy of calculations varied with learners’ skill level and problem difficulty. There was a clear contrast in the gesturing behavior and calculation performance of learners at different skill levels. Learners first mastered how to calculate using a physical abacus and later benefitted from using abacus gestures, answering more questions correctly when allowed to gesture compared to not gesturing. This suggested that learners acquired the ability to calculate using visualmotor spatial sequence, as the arrangement of abacus beads, followed by motorspatial sequence, as abacus gestures.
At each skill level, the differences between using a physical abacus, gestures, or no gestures also varied according to problem difficulty. The results indicated that as demands on working memory increased with problem difficulty, gestures assisted up to a point for mental arithmetic before learners resorted back to performing better on a physical abacus. Hand movement accuracy for especially intermediate and advanced learners also reflected motor learning. The difference in movement accuracy between the physical abacus and handsfree spontaneous gesture conditions showed a trend in increased movement accuracy following skill level; beginning learners had low movement and gesture accuracy while intermediate and advanced learners had high accuracy.
More specifically, beginning learners were able to perform calculations with a physical abacus even up to threedigit problems. However, they performed poorly in both handsfree and handsrestricted conditions to the point that at twodigit and threedigit problems, there was no significant difference between the two mental arithmetic conditions. This showed that beginners were able to correctly solve some difficult problems when able to see the arrangement of beads on a physical abacus, but did not benefit much from using gestures to manipulate an imaginary abacus for mental calculations. This pattern was also reflected in the poor accuracy of beginners’ hand movements. Movement accuracy was greatest with a physical abacus, especially for onedigit problems. But, under the handsfree mental arithmetic condition, gesture accuracy was equally poor for onedigit and twodigit problems, and nearly all inaccurate for threedigit problems. This clearly indicated that beginners had not yet learned how to calculate using gestures and still needed the aid of a physical abacus.
In contrast, gestures facilitated problem solving for intermediate learners in the handsfree condition, compared to both the physical abacus and handsrestricted conditions. The trend showed that onedigit problems were simple enough for intermediates to perform equally well in all conditions. At twodigit problems, intermediates could calculate just as well using gestures as with a physical abacus, but not when their hands were restricted. By threedigit problems the contrast was even clearer. Intermediates performed best with a physical abacus, indicating that intermediate learners’ ability to use gestures assisted only up this point. Yet notably, at threedigit problems, intermediates performed mental arithmetic significantly better when allowed to gesture compared to when their hands were restricted from moving. This clearly showed that intermediate learners had gained the ability to successfully use gestures as well as a physical abacus up to two digits, but not three digits. And, when the demands on working memory were highest at threedigit problems, gestures had a beneficial effect compared to not gesturing during mental arithmetic.
This beneficial effect of gestures also seemed to be related to movement accuracy. Overall, intermediates’ movement accuracy with gestures was almost as high as with a physical abacus. This trend continued for advanced learners, whose hand movement accuracy was just as high with spontaneous gestures as with a physical abacus. Interestingly, intermediates’ movements were significantly more accurate at twodigit problems compared to simpler onedigit problems or more difficult threedigit problems.
Studies of motorskill learning and automaticity have shown that novice and intermediate learners perform better under conditions for onlineattentional monitoring of their movements, while advanced learners perform better when explicit attentional control is prevented (
Advanced learners showed a mastery of mental arithmetic even without the use of gestures or a physical abacus. At two digits, advanced learners performed equally well using just gesture compared to using a physical abacus. In contrast to intermediate learners, advanced learners performed equally well at three digits in the handsfree and handsrestricted conditions. This indicated that advanced learners could use and maintain a mental representation of the abacus even without gesture. In contrast to beginning and intermediate learners, advanced learners gestural movements were highly accurate, regardless of problem difficulty. This indicated a higher degree of motor automaticity and internalization of the abacus representation for advanced learners compared to intermediate and beginning learners.
These contrasts among beginning, intermediate, and advanced learners in calculation performance and movement accuracy support the interpretation that abacus cothought gestures are learned as a motorskill that complements visualspatial mental representation. A growing body of research shows that motor and visual imagery are complementary processes (
Additional behavioral and fMRI studies have demonstrated that visualspatial and motorspatial sequences are acquired at different rates and skill levels.
Analogous to grouping strategies for visual working memory, motor sequences can be grouped or “chunked” as gestures.
It is possible that correct abacus gestures form chunked motor sequences representing arithmetic operations, thereby facilitating mental calculations. Once acquired as motorchunks, learners are then able execute combinations of these gestures in a series to perform more complex calculations. Such conceptual and motor chunking may reduce cognitive load. Skilled learners, who have acquired abacus gestures, thus need only to decide on which gesture to execute, given the arrangement of beads. This reduces working memory load when calculating because changing the arrangement of beads is executed as a motor sequence. While it takes time to learn how to use gestures to do mental arithmetic without the visual assistance of a physical abacus, advanced learners can execute the calculation easily once they have acquired the learned motorsequence.
Previous fMRI studies of abacus mental arithmetic have shown greater activation in nonexperts compared to experts of frontalsubcortical areas related to the global workspace of executive function (
Recent studies of abacus mental arithmetic and task switching have found that abacus training also improves higherorder math abilities beyond basic arithmetic, multiplication, and division. Longterm learners perform significantly better than untrained peers on more abstract tasks including algebraic number filling (e.g., 4+_ = 3 + 7), number sequence recognition, numerical working memory, and visualspatial counting and matching (
Abacus cothought gestures have a clear beneficial effect for maintaining a mental representation of the abacus while performing mental arithmetic. These gestures are learned as specific movements using the index fingers and thumbs for moving abacus beads according to algorithms for complements of 5 and 10. Learners first acquire a basic skill in using a physical abacus and then acquire proficiency in using abacus gestures. The results indicate that this beneficial effect and accuracy of abacus gestures is related to motor learning. Beginners benefit little from using abacus gestures and their movement accuracy is poor. Intermediates perform mental arithmetic better when allowed to spontaneously gesture compared to when their hands are restricted. According the Gesture as Simulated Action theory, such spontaneous gestures are used when the demands of working memory reach a threshold. Advanced learners’ mental abacus score and gesture accuracy were comparatively high, regardless of whether they gestured or not. This indicates that they had automated the motor programs of abacus gestures. Such automated motor programs can be executed with little conscious effort or demand on working memory. These results are consistent with previous findings on mental arithmetic that found that learners at different skill levels improved in their use of visual strategies. Moreover, our findings suggest that abacus gestures act as motor programs that complement such visualmotor representation. This interpretation is supported by behavioral and neurophysiological studies which indicate that visualspatial and motorspatial learning are two complementary systems.
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Special thanks to Mr. Tay and all the teachers at CMA Singapore without whose dedication this study would not have been possible. Thanks also to Dr. Ho Yan Yin and Ms. Sally Kwon.
The Supplementary Material for this article can be found online at: