INFLUENCE of BILINGUALISM on SIMPLE ARITHMETIC by Towhid Nishat a Thesis Submitted to the Faculty of the Charles E. Schmidt Coll

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INFLUENCE OF BILINGUALISM ON SIMPLE ARITHMETIC

Towhid Nishat

A Thesis Submitted to the Faculty of

The Charles E. Schmidt College of Science

In Partial Fulfillment of the Requirements for the Degree of

Master of Arts in Psychology

Florida Atlantic University

Boca Raton, FL

May 2015

ACKNOWLEDGEMENTS

The author wishes to express sincere gratitude to his committee members for all of their guidance and support, and special thanks to my advisor for her persistence, patience and encouragements during the writing of this manuscript. The author is very grateful for all the time and effort spent by Giselle Perez and Katherine Diaz to conduct the study and provide operational support throughout. Last but not least, the author wishes to thank his lab mates for all their support.

ABSTRACT

Author: Towhid Nishat

Title: Influence Of Bilingualism On Simple Arithmetic

Institution: Florida Atlantic University

Thesis Advisor: Dr. Monica Rosselli

Degree: Master of Arts

Year: 2015

It has been widely hypothesized that while doing arithmetic, individuals use two distinct routes for phonological output. A direct route is used for exact arithmetic which is language dependent, while an indirect route is used during arithmetic approximation and thought to be language independent. The arithmetic double route has been incorporated on the triple- code model that consists of visual arabic code for identifying strings of digits, magnitude code for knowledge in numeral quantities, and verbal code for rote arithmetic fact. Our goal is to investigate whether language experience has an effect on the processing of exact/approximation math using bilingual participants who have access to two languages, using a theoretical arithmetic processing model, which has been validated across many studies. We have measured the two groups

(monolinguals/bilinguals) processing speed for completing the two tasks

(Exact/Approximation) in two codes (Arabic digit/Verbal). We hypothesized a faster reaction time in exact arithmetic task in compared to approximation in accordance with

v the triple-code model. We also expected a main effect for the task (Exact vs.

Approximation) independent of the input code when the stimulus was presented in either

Arabic digit and/or verbal codes. Our results show exact arithmetic is faster than approximation of arithmetic facts in all codes supporting earlier theories. Also, there was no significant difference in processing speed between monolinguals and bilinguals when performing the arithmetic task in either Arabic and/or verbal codes. In addition, our investigation suggests a modification to the triple-code model when interpreting arithmetic facts in verbal code due to interference of two languages with bilingual participants. Additions to the model can be suggested when the stimulus is expressed in verbal code for visual identification, which may cause interference in bilinguals leading to a first language advantage due to language experience.

INFLUENCE OF BILINGUALISM ON SIMPLE ARTIHEMETIC

List of Tables ...... viii List of Figures ...... ix Introduction ...... 1 Perception and processing of numbers ...... 3 A bilingual’s advantage to task switching ...... 8 Hypothesis being explored by this investigation ...... 15 Methods...... 18 Participants ...... 18 Procedures ...... 19 Materials ...... 20 Vigilance task ...... 21 Arithmetic tasks ...... 22 Result ...... 25 Discussion ...... 39 References ...... 45

vii

LIST OF TABLES

Table 1. Participants Demographics ...... 19

Table 2. Means and standard deviation of Raw Score and Post Vigilance for each tas .. 27

Table 3.Correlations between the verbal code and self rated language proficiency for

bilinguals...... 34

Table 4 . Correlations between the verbal code and self rated language proficiency

for monolinguals ...... 35

Table 5. Correlations between the verbal code and the total self rated language

proficiency for monolinguals ...... 37

Table 6 . Correlations between the verbal code and the total self rated language

proficiency for bilinguals ...... 37

Table 7. Linear Regression for task and total proficiency...... 38

Table 8. Language in which daily task is completed by bilinguals participants ...... 42

viii

LIST OF FIGURES

Figure 1. Mean differences on post vigilance reaction time between monolinguals

and bilinguals on arithmetic tasks in Arabic and in English verbal code...... 28

Figure 2. Post vigilance reaction time for bilinguals on the verbal arithmetic tasks ...... 29

INTRODUCTION

Language has aided in the communication of civilization from the beginning of time. Language comes in various forms ranging from hieroglyphics, sign language, and countless numbers of spoken languages. Spoken language is essential in our society for various cognitive abilities we as individuals use in our daily life. One such cognitive task is the ability to do various arithmetic problem-solving involving many different strategies, and representation. As diversity increases in modern society, more individuals use more than one language in their daily communication. It has been estimated that more than half of the world population, close to 60%, is bilingual or multilingual (Baker,

2001). With more individuals using two or more languages in their daily life, one can assume that certain strategies bilinguals use to successfully complete tasks differ from monolinguals. The culture and ethnic diversity the bilingual speaker has access to while in a particular environment may causes them to have greater cognitive flexibility to perform the tasks at hand being more advantageous then monolinguals.

One cognitive area in which bilinguals may behave differently from monolinguals is in solving arithmetic problems. In our investigation, we are interested in seeing how fast bilinguals manipulate numbers in comparison to monolinguals on simple arithmetic facts using a triple-code model for numeral processing (Dehaene & Cohen, 2004). This particular study will only concentrate on basic arithmetic facts to analyze how English speaking monolingual and English/Spanish bilinguals participants process numbers when 1 such numeral facts are written in Arabic digit and in verbal code such as English and

Spanish. How numerals facts are interpreted by bilinguals in comparison to monolinguals would provide us with a better understanding of how speaking more than one language influences the speed of solving arithmetical facts, as the processing speed for monolinguals would be used as a baseline for comparison with the bilingual participants.

PERCEPTION AND PROCESSING OF NUMBERS

Dehaene and Cohen (1997) proposed a ”triple-code model” of the cognitive and neuro-anatomical architecture for number processing. This model supports the dissociation between two arithmetic tasks such as exact arithmetic dependent on the area of the brain responsible for language in comparison to approximation of arithmetic facts relying on the visuo-spatial skills. The triple-code model includes visual arabic code, a magnitude code, and a verbal code (Dehaene, & Cohen, 1997). Different brain areas process each code. Prior to computation, the visual Arabic code is processed in the left and right inferior ventral occipito-temporal area, where the numbers are represented as an identified string of digits. The magnitude code, or analogical quantity, is associated with the left and right parietal lobes where numbers are represented as distribution of activation on an oriented number line. The verbal code for numbers is sub served by the left-hemisphere perisylvian area where numbers are represented as a parsed sequence of words such as “2” represented as “Two.” According to the model, a direct route and an indirect route can be accessed during multiplication and addition of arithmetic facts. The components of the direct route can be activated in a linguistically mediated operation such as visual identification (2 x 4), visuo-verbal transcoding (“two times four”), and verbal sequence completion (“two times four, eight”). In accordance with the triple-code model, indirect route can be employed during quantity comparison and subtraction, depending on manipulation over the “analogue magnitude,” which is responsible for

3 representing numbers as language independent mental magnitude (Dehaene, & Cohen,

1997). According to the model, direct route for arithmetic processing is used during performance of exact arithmetic and theorized to have a faster reaction time than performing approximation of arithmetic facts which employs the indirect route. This project aims to understand arithmetic processing in participants with different language experiences using the triple code model. It explores how English monolinguals and

English/Spanish speaking bilinguals can interpret simple arithmetic sums in exact and approximation. Thus according to the model, we should expect faster reaction times for all participants (monolingual and bilinguals) when arithmetic tasks are completed for exact arithmetic compared to approximation tasks due to the reliance of the former on direct route and the latter on the indirect route; this pattern will be observed whether the stimulus is presented in Arabic digit or Verbal code. To observe between group differences, English speaking monolinguals’ performance on the arithmetic task would be used as a baseline to observe differences on bilinguals to investigate if having access to two languages will be beneficial while performing a simple arithmetic task. As an exploratory measure, within group differences would be observed to investigate how language experience can play a role in arithmetic processes when completed in verbal code.

One question proposed by Dehaene and colleagues (1999) was “Does human capacity for mathematics intuition depend on linguistic competence or a visuo-spatial representation?” The two aspects of arithmetic discussed by the authors are exact calculation, which can be seen as language dependent, and approximation, which can rely on a non-verbal, visuo-spatial cerebral network. In the study, Russian-English bilinguals 4 were tested in approximation and in exact arithmetic. They were taught a set of sums of two digit numbers in one of their two languages. In the exact condition, the subject chose the correct sum from two numerically close answers, while in the approximation condition the subjects were told to estimate the sum and choose the closest response. In addition, after training the subjects’ response time for solving trained problems and novel problems, both conditions were tested in their two languages. The results from the study indicated subjects performed faster in the language they were taught than in the untrained language, demonstrating when individuals were trained in a particular arithmetic problem, the stored information was in a language-specific format and showed a language-switching cost due to required internal translation of the arithmetic problem

(Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). In contrast, in the condition with approximation sums, the subjects’ performance was equivalent in the two languages, providing evidence that knowledge acquired by exposure to approximate problems are stored in a language-independent form. According to the authors, exact arithmetic puts emphasis on language–specific representation and relies on the left inferior frontal circuit; in comparison, approximation does not show dependency in language and relies primarily on a quantity representation implemented in a visuo-spatial network of the left and right parietal lobe..

Exact and Approximate Arithmetic in an Amazonian Indigene group

To support the claim that approximations of arithmetic facts are language independent,

(Pica et al. 2004) have looked at indigenous groups who have learned a language without numeral facts above 4, while also lacking the ability to do simple arithmetic computations. In this indigenous group, individuals were able to estimate large values 5 above 4 using visuo-spatial skills, close to or better than a population of French controls, therefore it can be deduced that approximation of arithmetic fact is language independent.

This supports the theory claimed by the triple code model that approximation is language independent. It has been proposed by Pica and colleagues (2004) that the basic aspect of numerical cognition depends crucially on language ranging from vocabulary of counting words, to syntax of arithmetic facts. This article argued numerical abilities involve two different mechanisms, one for a small set of 4 or less and another for a larger set, such as the word “one” representing one object and “two” representing two objects. Sets of numerals from 1-4 are consistent and representative at each successive stage in early learning, with values increasing as knowledge is gained of numeral facts. Language is crucial with understanding the various terminologies dealing with arithmetic. At this particular time it would be interesting to observe groups of individuals such as the

Munduruku and Piraha tribes of the Amazonian Indians. These tribes used numeral facts such as small sets less than 4, while lacking number names greater than 5. In addition, if the group observed any amount of objects greater than 5, they computed the answer by equating it to their hands and feet (Pica et al., 2004). For example, if they saw 12 dots on the screen, they would respond as two hands and two toes from their feet. The purpose of this study was to examine numeral competence in situations in which the language of numbers is either absent or reduced. The first task explored by the authors was showing 1

– 15 dots in a randomized order to the participants, then asking how the numbers were being represented in their native language. The native language being investigated was

Munduruki, a language of the Tupi family, spoken by approximately 7,000 people living in the autonomous territory in the Para state of Brazil. The findings were interesting

6 because above the number 5 there was little consistency in language use and higher numbers were represented by words translating to “some” or “many.” For example, when the subjects were asked to represent 13 dots, they represented them by “all the fingers of the hand and some more.” The study was compiled of 55 native speakers of Munduruki in a computerized battery with ten native speakers of French as the control. When the

Munduruku group was tested with various non-verbal tasks using numerosity, they succeeded on non-verbal number tasks that used displays that represented as large as 80

(Pica et al., 2004). The key claim made by the author was that language is necessary for mental representation and manipulation for numerosity greater than 4. The task required each participant to compare sets of dots, approximate each set’s addition, and compare their results. In addition, two other tasks in which they solved for the exact subtraction and pointed to the result, and in the other named the result of the exact subtraction stimuli. The following group consisted of participants in which some were monolingual

Munduruku while others were bilingual (Portuguese) and was compared to French speaking individuals. Bilingual adults and children performed like the monolingual speakers. The Munduruku continued to deploy approximate representations in a task that the French controls easily resolved by exact calculation.(Gelman, & Butterworth, 2005).

In all of the comparison tasks presented to both groups the Munduruku performance was close to that of controls, however at times when exact arithmetic was needed, their performance significantly dropped in comparison to that of the controls,’ concluding that approximation can be done through various visuo-spatial processes (being language independent in comparison to exact arithmetic being dependent to language for a correct response).

A BILINGUAL’S ADVANTAGE TO TASK SWITCHING

The triple-code model, when numeral values are recognized and the individuals are asked to calculate a successful response to either exact or approximating arithmetic facts, may lead to minor interference while calculating exact/approximation sums in verbal/Arabic code due to double dissociation not consciously recognized. When given a cognitive task, an individual may employ skills such as updating of working memory, inhibiting distracters or responses, and shifting between mental sets. Bilinguals may differ from monolinguals in certain cognitive tasks due to constant shifting between mental sets depending on their environment and the audience they are in communication with. Such executive control can also be observed when bilinguals are asked to perform a certain task whether using their primary or secondary language. In bilinguals, advantage in executive control is assumed to stem from bilinguals constant need to manage and monitor their two languages (Prior & Macwhinney, 2010). The demand on language switching in bilingual speakers, according to Prior & Macwhinney, may be parallel in comparison to task switching in which both paradigms rely on executive function of mental shifting. Prior & Macwhinney investigated possible bilingual advantage in shifting between mental sets by using a non-linguistic task-switching paradigm and found a pronounced reduction in switching cost for bilinguals. Using a participants pool of forty-five monolingual (32 female) age 18.7 years (SD=.9) and forty-seven bilinguals (27 female) age 19.5 years (SD=1.5), each participant was asked to complete the Verbal

8 component of the SAT and the Peabody Picture Vocabulary Test (PPVT-III) to test for verbal ability. In order to test for task switching, graphic task cues were used such as color gradient and shapes. In each case the “red” response was assigned to the index finger and the “green” response was assigned to the middle finger. Similarly, the “circle” response was assigned to the index finger and the “triangle” response was assigned to the middle finger. The response key for the color and the shape was labeled respectably so as not to have any confusion. In addition to task switching, participants had to complete an

Operation span task, where the participants solved mathematical expressions while maintaining sets of English words in their memory in order to compare monolingual and bilinguals in working memory. Switching costs are defined as the differences in performance on switch trials as opposed to non switch trials. The author reported interaction between trial type and language group was significant in the RT but not the accuracy. The author also reported that this interaction is driven by the fact that both language groups performed identically on non-switch trials, however bilinguals were much faster than monolinguals on the switch trials. Thus, bilinguals incurred a much lower switching cost than monolinguals (Prior & Macwhinney, 2010). In bilinguals, enhanced executive function, selection of the appropriate language to perform a task, and rejection of competition and interference from the other language to reach a successful phonological output, has been observed in many investigations. In our current study we are investigating exact and approximation of arithmetic sums written in Arabic digit and

Verbal code. In accordance with this article if bilinguals have an executive advantage over monolinguals and show a much lower switching cost, then we will be able to

9 observe differences in scores between the groups when performing tasks such as exact sums written in verbal code.

Dependence of numerical cognition on language was a focus on a bilingual fMRI study done by Venkatraman and colleagues (2006). The study manipulated language in order to study its effect on exact and approximate arithmetic processing. This study involved twenty neurologically normal right handed, English-Chinese bilinguals. The authors chose two different tasks that were neither commonly taught in school nor used in everyday life. The tasks were base-7 addition tasks involving exact arithmetic and a percentage value estimation task involving approximation. In this design, the authors trained the participants in both tasks, changing the language of training between tasks.

Half of the participants were trained on the approximate task in English and the exact task in Chinese. The other half was trained in the approximation task in Chinese and exact task in English. The authors predicted that for the base-7 addition problems participants would acquire new rules for performing exact calculation leading to engagement of verbal processing. For the approximation task, the authors predicted that representation underlying the processing of these problems would be largely language independent and would not recruit cortical areas typically involved in language processing. A two-way repeated measures ANOVA was computed. Analysis of the reaction time data revealed a significant main effect of training, as well as a main effect of the type of calculation. Thus, the authors reported that the calculation condition reaction times were faster for the trained than the untrained language, with no significant difference in accuracy. The language switching effect for exact arithmetic was most prominent in the bilateral frontal region and in the left inferior parietal lobule. From the

10 scope of this study, the author suggests a greater involvement of language –related neural circuits during exact number processing. When the subjects were being trained in the particular task, they stored the knowledge in the language in which they were trained .

When confronted with the problem in which they were untrained, they switched to the language in which they were trained to find a proper solution strategy to solve the problem. A language switching effect for approximation percentage calculations was reported by the author to be seen predominantly in the bilateral posterior parietal area and the left prefrontal cortex which suggested a greater processing demand on visuo-spatial circuits during estimation and lesser activation of language related neural circuits

(Venkatraman, Siong, Chee, & Ansari, 2006). In conclusion, switching language for a successful completion may depend on how proficient the individuals are in a particular language and their understanding and comfort in interpreting numeral values in such language.

The Relationship between Performance on Math and Language Proficiency

It has been implied that bilinguals who are highly proficient in one of their learned languages tend to do better in solving word problems in such language of proficiency than in their other language (Kempert, Saalbach, & Hardy, 2011). One study was conducted on Irish (Gaeilge)/English bilinguals on their proficiency in both languages and then compared to their performance on mathematical word problems on the perspective languages. Investigators tested English monolinguals and Gaeilge and

English bilingual school children. Bilingual participants were required to take a language proficiency test in English and in Gaeilge prior to their participation in the study. They

11 were able to classify the bilinguals into four groups: (1) those who were highly proficient in both Gaeilge and English (high/high), (2) those who were highly proficient in only

Gaeilge with poor English (high/low), (3) those who were poor in Gaeilge and highly proficient in only English (low/high), and (4) those who were poorly proficient in both languages (low/low). The results suggest that those who are highly proficient in two languages perform better solving the word problems than those who are either highly proficient in one language or poorly proficient in both languages. Results also suggest that the high/high group outperformed the monolingual group suggesting that bilingualism is beneficial in terms of mathematics. Investigators suggest that developing mathematical literacy through the medium of Gaeilge at the primary level (first eight years of education) will augment the transfer to the English medium mathematics at the second level (six years after the primary level). Investigators also suggest that proficiency of bilinguals in English and Gaeilge should be assessed during the beginning of second or third level (college) education (Riordain and O’Donoghue, 2009). This will ensure that students will have a high proficiency in one language which will help them learn the new language, ensuring that mathematical skills will be enhanced. Although this was found, little is still known about how the level of language proficiency of bilinguals correlates with processing speed of exact math tasks. In accordance with the triple code model exact arithmetic is considered language dependent in comparison to approximation which is considered to be language independent. If this is true, then a correlation between language proficiency and the RTs in the exact arithmetic tasks should be expected, whereas no associations between language proficiency and RTs should be expected in the performance of approximation tasks.

In summary, many researchers using the triple-code model designed by Dehaene supported the double dissociation in the brain during arithmetic processing. According to the triple code model exact arithmetic is considered to be language dependent and rely on language specific areas of the brain, whereas approximation is considered to be language independent relying on parietal region employing visuo-spatial skills for a successful response. In addition, when looking at arithmetic from a language perspective, groups with very limited language can use visuo-spatial strategies to approximate answers very close to individuals with developed language, which supports the model that approximation can be language independent. In addition, through imaging studies it was supported that double dissociations do exist during arithmetic processing and that computing exact and approximation of sums are independent of each other. It was also observed that when bilinguals were asked to answer a particular question, they would have a first language advantage, and performs better in the language of training or proficiency than the untrained language in arithmetic. In addition, it can be deduced that being proficient in two languages can be quite beneficial when it comes to arithmetic problem solving. Our goal is to investigate using the triple code model for arithmetic processing whether language experience has an effect on the processing of exact/approximation arithmetic using English/Spanish speaking bilingual participants who have access to two languages, and whether such processing is affected when the arithmetic calculation is perceived in double codes, such as Arabic digit code or as

English/Spanish verbal Code. Until now, the triple-code model was tested repeatedly by many researchers using Arabic digit code, so it will be interesting to observe how an

13 individual processes arithmetic sums when the paradigm is transcribed in verbal code while having access to two distinct languages.

HYPOTHESIS BEING EXPLORED BY THIS INVESTIGATION

Using a 2x2x2 mixed design, the present study will explore weather arithmetic is language dependent or independent comparing monolingual and bilingual participants.

This study will test their response times in tasks (Exact and Approximation) requiring the use of double code for calculations (Arabic digit and verbal codes). Two within subjects conditions will be included; exact and approximation calculations in Arabic digit code and verbal code; the between subject condition will be language group (bilinguals vs monolingual)

If exact arithmetic is considered language dependent, we should be able to observe a faster reaction time for monolinguals and English/Spanish bilinguals across all codes (Arabic digit, English verbal and Spanish verbal) in comparison to approximating of arithmetic facts. A main effect in task is expected, and such faster reaction time can support previous investigations of the triple code model that exact arithmetic is language dependent in comparison to language approximation which been theorized to be language independent due to the implementation of visuo-spatial skills. This would corroborate previous findings using the triple code model with two additional codes to examine differences between monolingual and bilinguals while observing what influence the different codes may have on solving simple arithmetic sums. We would expect the bilingual group to be slower in reaction time than monolinguals when performing the arithmetic task presented in verbal code due to possible interference between two 15 languages. An interaction is expected for monolingual participants being faster on performance of exact arithmetic when stimulus is presented in English verbal code.

However, we do not expect to see differences between groups when calculating in Arabic digit code.

In addition, a 2x2 between subjects design with two conditions with two levels each will be included using the bilingual sample; exact and approximation calculations in

English verbal code, as well as Spanish to observe within subject difference in the bilingual group. The purpose of using a bilingual group is to observe how language experience can influence arithmetic processing when presented in verbal code. We would hypothesize a main effect for task observing a faster reaction time for exact arithmetic, in addition to main effect for code being faster on verbal code with greater language experience. An interaction is expected for exact arithmetic due to it being theorized to be language dependent when presented in verbal code of the participant’s language of greater experience.

Self-rated language proficiency in English for monolingual and English/Spanish for bilinguals required each participants to rate themselves (1= virtually nothing to 5 =

Excellent) on how they read, write speak and understand English and/or Spanish.

Correlation between self- rated language proficiency, and the RTs from the verbal code in the exact arithmetic tasks should be expected whereas no associations between language proficiency and RTs should be expected in the performance of approximation tasks. Since approximation of arithmetic facts relies on visuo-spatial skills and is theorized to be language independent, then no association is expected between how proficient the

16 participants rate themselves to be in either language and the processing time in such task.

It should be noted at this time, that participants were not given a formal test to observe proficiency in one or both of their languages, and answers can be used from the questionnaire as a valid way to understand perceived proficiency of an individual.

METHODS Participants

The participants were recruited from the FAU subject pool as well as students enrolled in psychology classes on the Davie campus. The exclusionary factor for this study was to exclude participants who were unable to comprehend the English language properly thus hindering their understanding of the instruction of the task being investigated. In addition, we excluded all participants who spoke languages in addition to

Spanish and English for the bilingual population for the purpose of comparing only

English speaking monolinguals with English/Spanish speaking bilinguals. The records collected from the monolingual population would be compared to the records collected from the bilingual population in order to understand any differences in the scores, as performance for the monolingual population can be used as the baseline for the language, which for the purpose of this study would be English. A total of 164 individuals participated in our investigation,117 were females (100 English speaking monolinguals and 64 English/Spanish speaking bilinguals) within the ages of 18 to 34 (M=23.08,

SD=3.22 ). Further demographic of the participants are provided in Table 1.

Table 1. Participants Demographics Participants Monolingual Bilingual Gender Males 28 (28 %) 19 (29.7 %) Females 72 (72 %) 45 (70.3 %) Age Mean = 23.34 (+/- 2.95) Mean = 22.33 (+/- 3.06) Years in the USA Mean = 21.04 (+/- 7.05) Mean = 17.34 (+/- 6.20) European-American 34 (34 %) 2 (3.13 %) Hispanic 10 (10 %) 62 (96.9 %) Ethnic Afro-Caribbean 14 (14 %) 0 Background African-American 11 (11 %) 0 Asian 12 (12 %) 0 Native-American 0 (0 %) 0 Others 19 (19 %) 0 Number in years of education * Mean = 16.54 (+/- 1.85) Mean = 16.60 (+/- 2.78) High School 10 (10 %) 10 (15.63 %) Diploma Associates Degree Highest Degree 68 (68 %) 47 (73.44 %) Attained Bachelors Degree 21 (21 %) 6 (9.38 %) Masters Degree 1 (1 %) 1 (1.56 %) Middle School 0 0 Math Algebra 1 2 (2 %) 0 Highest Math Geometry 1 (1 %) 0 level completed Algebra 2 2 (2 %) 7 (10.94 %) Statistics 39 (39 %) 16 (25 %) Calculus or 56 (56 %) 41 (64.06 %) above * From kindergarten to 12 grade was counted as 13 years of education, while any education beyond that was greater than 13.

Procedures

All participants consented to the study prior to receiving a Mini Mental State

Examination to screen for any cognitive impairment. In addition, all participants were given a questionnaire to better understand their demographics as well as questions in which participants self-rated their proficiency in English (monolingual) and their proficiency in English and in Spanish (bilingual). All participants rated themselves in the questionnaire from 1-5 (1=Virtually nothing to 5= excellent) on how well they read, write, speak and understand in English. While the bilingual participants reported their proficiency on how well they understand, speak read and write in English as well as

Spanish. Following the questionnaire, the participants performed the computerized arithmetic task, which was programmed in to DirectRT that asked in one trial to give the exact response to a simple arithmetic sum and in another trial to approximate the arithmetic sums. The arithmetic tasks was tested in both Arabic digit code and English verbal code for all participants, while only bilinguals had an extra block with exact and approximation task written in Spanish verbal code. The experimenter, who was fluent in

Spanish and in English, instructed the bilingual group in Spanish, prior to beginning the

Spanish verbal tasks.

Materials

The investigator programmed the arithmetic task explored in this investigation into DirectRT software created by Empirisoft INC. The task itself was sent to the investigator by Dehaene (1999), which included the task of exact/approximation of arithmetic fact in Arabic code. The investigators of this study transcribed the task into

English verbal code as well as Spanish verbal code as an addition to the paradigm. The purpose of transcribing the same task in English and in Spanish was to test the triple-code model when the arithmetic sums are written in verbal code, and what effect it may have on the bilingual participants with interference between two languages. A total of 40 trials per block were randomized in each block, so the participants being tested in each code and task would never be exposed to arithmetic facts in any specific order. In addition, to better understand the processing speed for each arithmetic task, a vigilance task was added to the paradigm and the average time it took the participants to complete the vigilance task was subtracted from the average time it took the participants to complete

20 each condition. Processing speed of only correct responses were used for our analysis from each arithmetic task.

Two types of task were required from each participant to test our hypothesis: (1)

Vigilance task which required participants to respond to symbols they had observed as stimulus (Observe and Respond), and (2) an Arithmetic task which required participants to observe two single digits on the screen then either calculate an exact response or approximate the sums by pressing the appropriate key on the keyboard (Observe,

Calculate and then Respond). For the purpose of understanding the processing speed for arithmetic exact and/or approximation calculation in various codes, participant’s average processing time for the vigilance tasks (in milliseconds) was subtracted from the arithmetic task (in milliseconds).This afforded a better understanding of the average processing time needed for the participant to complete the arithmetic task. In the analysis and in all figure and tables, “PV” written after each task represents post vigilance scores when average vigilance time were subtracted from the arithmetic task. Incorrect responses to any stimuli were excluded from the analysis.

Vigilance task

For the symbol vigilance task, participants fixated on the center of the computer screen, then observed an asterisk (*) which appeared for 1,000 ms. After the asterisk disappeared, the stimulus appeared on the screen such as (♠ ☺) for 600 ms. After the stimulus disappeared, research participants observed (☺ ♪) and had to press either, the left alt key or right alt key, corresponding to the symbol they saw as a stimulus. A new stimulus was presented after each asterisk was observed. In addition to symbol vigilance,

21 an added vigilance task consisted of letters observed for stimulus, such as (F J).

Participants were asked to respond to the letter they previously saw from choices such as

(J D). For all vigilance and arithmetic tasks, the asterisk appeared on the screen for

1,000 ms for fixation, while the stimulus was presented for 600 ms. To measure reaction time, no limit was set for the response screen. Thus, when participants pressed the key, the reaction time was recorded and would execute the next trial. In addition, the response for the vigilance task was randomized so as not to observe a Simon effect where reactions would be more accurate if the stimulus and the response were in the same relative location.

Arithmetic tasks

All participants were required to complete the arithmetic task in Arabic digit and

English verbal code. Only bilinguals were required to complete an extra arithmetic task in

Spanish verbal code for exact and approximation of arithmetic sums. Orders of each task were randomized for participants to avoid order effect. Examples for each task listed were completed in randomized order. Each block (arithmetic tasks) had a total of 40 trials

(questions) the participants needed to complete successfully, which later was analyzed with only the correct response.

Arabic Exact: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (3 + 4) for 600 ms. The participants responded by pressing the left alt key or right alt key corresponding to the answer (10 7).

Arabic Approximation: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (3 + 4) for 600 ms.

The participants were asked to approximate the response by pressing the left alt key or right alt key corresponding to the answer (6 10).

English Exact: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (Three + Four) for 600 ms. The participants responded by pressing the left alt key or right alt key corresponding to the answer (Ten Seven).

English Approximation: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (Three + Four) for

600 ms. The participants were asked to approximate the response by pressing the left alt key or right alt key corresponding to the answer (Six Ten).

Spanish Exact: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (Tres + Cuatro) for 600 ms. The participants responded by pressing the left alt key or right alt key corresponding to the answer (Diez Siete).

Spanish Approximation: Participants fixated on the computer screen, then observed an asterisk (*) appear for 1,000 ms, after which the stimulus appeared (Tres + Cuatro) for

600 ms. The participants were asked to approximate the response by pressing the left alt key or right alt key corresponding to the answer (Seis Diez).

Our goal was to observe the bilinguals and their reaction times on double code tasks involving arithmetic problems. With such paradigm consisting of exact and approximation tasks, we can observe language dependency and its effect in calculations presented in two codes and how it may be related to bilinguals as well as monolinguals.

The paradigm was designed to observe between group differences on the language condition as well as within group differences on the exact and approximation calculation tasks in both codes. The objective was to examine if language experience has an effect on processing of arithmetic facts when the individual has access to two languages. Finally, all participants had a brief interview with the experimenter discussing how they have completed the tasks and if they employed any strategies during the verbal code of the arithmetic tasks

RESULT

Independent variable for this investigation was the arithmetic task (Exact and

Approximation) written in Arabic digit code as well as newly transcribed verbal code

(English and Spanish). Processing speed, recorded as reaction time in milliseconds, on each task written in each code was analyzed as the dependent variable using 2x2x2 repeated measures ANOVA to observe any differences between and within the groups

(Monolingual and Bilingual). In addition, to understand within subject difference and how having access to two distinct languages affect calculation, a 2x2 repeated measures

ANOVA was conducted to analyze the processing speed for bilingual participant’s performance in the calculation task (Exact and Approximation) written in English and

Spanish verbal code. Furthermore, to explore language experience and its affect on calculation of arithmetic sums, we performed a correlation with the participants self-rated proficiency (1= virtually nothing to 5= excellent) on how they understand, speak, read and write in English and in Spanish with their performance on the arithmetic task presented in each code.

Our investigation explored the idea if arithmetic is language dependent and/or independent utilizing the triple-code model for arithmetic processing. Performance on tasks presented in Arabic digit code as well as verbal code (English and Spanish) was observed from monolingual and bilingual participants to explore how language experience may affect performance on calculation. In order for us to better understand the 25 processing time for calculation of each task (Exact vs. Approximation) presented in each code, we analyzed the subtracted reaction time in milliseconds of the participants average vigilance task from their arithmetic tasks. We found no significant between group differences on the vigilance task (Average vigilance reaction time for English speaking monolinguals (Mean = 654.59 ms (+/- 155.51)) and English/Spanish speaking bilinguals

(Mean = 655.12 ms (+/- 131.68)).

. Means and standard deviation of Raw Score and Post Vigilance for tas and for each Vigilance Post Score deviation and Raw of Means . standard

2 Table

Figure 1. Mean differences on post vigilance reaction time between monolinguals and bilinguals on arithmetic tasks in Arabic and in English verbal code.

Figure 2. Post vigilance reaction time for bilinguals on the verbal arithmetic tasks

We expected, independent of codes, a faster reaction time for exact arithmetic, thought to be language dependent according to triple-code model for arithmetic processing. A 2x2x2 repeated measures analysis of variance was conducted to observe differences in processing speed of task (Exact and Approximation), presented in codes

(Arabic digit and English verbal) for each group (monolingual and bilingual), and. The

ANOVA showed significant main effects for code (Λ (Wilks’ Lambda) = 0.265, F (1,

162) = 450.02, p < .001), also for task (Λ = 0.729, F (1, 162) = 60.075, p<.001.).

Participants were significantly faster while computing arithmetic sums in Arabic digit code (Total mean reaction time for Exact 38.09 ms (SD = 249.99) and Approximation

156.00 ms (SD = 233.17)) in comparison to English verbal (Total mean reaction time for

Exact 576.63 ms (SD = 442.40) and Approximation 720.26 ms (SD = 470.87)). In addition, processing speed for exact arithmetic was significantly faster than approximation of arithmetic sums for both groups as reaction time for Arabic exact

(Monolingual 18.11 ms (SD = 246.72), Bilinguals 69.32 ms (SD = 253.80)) and English verbal exact (Monolingual 534.20 ms (SD = 443.94), Bilinguals 642.92 ms (SD =

435.15)) was significantly faster than Arabic approximation (Monolingual 131.67 ms (SD

= 231.93), Bilinguals 194.01 ms (SD = 231.79)) and English Verbal Approximation

(Monolingual 716.56 ms (SD = 525.70), Bilinguals 726.05 ms (SD = 373.24)). The

ANOVA did not show an interaction for Code x Group (Λ = 1, F (1, 162) = .002, n.s.),

Task x Group (Λ = .989, F (1, 162) = 1.837, n.s.) and Code x Task (Λ = .998, F (1, 162)

= .308, n.s.). The average processing speed for each arithmetic task per group is listed in

Table 2, and it supports that exact arithmetic is faster than approximation of arithmetic facts regardless of the code it is written in. In order to observe how language experience may influence arithmetic processing in verbal code, bilingual participants were asked to complete both task in English and Spanish verbal code.

A 2x2 repeated measures analysis of variance was conducted to observe within subject difference in bilinguals and their performance on task (Exact and Approximation) written in two codes (English verbal and Spanish verbal). The ANOVA showed significant main effects for code (Λ (Wilks’ Lambda) = 0.734, F (1, 63) = 22.790, p <

.001), and task (Λ = 0.761, F (1, 63) = 19.794, p<.001.); Bilinguals performed better in

English verbal (Mean reaction time for Exact (642.92 ms (SD = 435.15)) and

Approximation (726.05 ms (SD = 373.24)) tasks in comparison to Spanish verbal tasks 30

(Mean reaction time for Exact (786.94 ms (SD = 473.71)) and Approximation (968.05 ms

(SD = 517.48)). Also, bilinguals in both codes performed significantly faster in exact arithmetic rather than approximation (Table 2 & Figure 2). An interaction was observed in the bilingual group as they performed English verbal task faster than compared to

Spanish while in exact arithmetic; Task x Code interaction was significant (Λ = .926, F

(1, 63) = 5.024, p<.05).

We were concerned if the language proficiency influenced such differences in performance in English verbal tasks compared to Spanish in the bilingual group. We investigated this phenomenon further by analyzing the data with two covariates on how the bilingual rated themselves in their proficiency in reading English (English read) and

Spanish (Spanish Read). Result from such analysis showed that the covariates interacted with Code (stimuli in English or Spanish), Code x English Read (Λ (Wilks’ Lambda) =

0.846, F (1, 61) = 11.11, p < .001), also for Code X Spanish Read (Λ = 0.688, F (1, 61) =

27.63, p<.001) and not with Task (Exact or Approximation), Task x English Read (Λ =

0.954, F (1, 61) = 2.96, n.s.), also for Task X Spanish Read (Λ = 0.958, F (1, 61) = 2.67, n.s.). From this analysis, we can state that when individuals rated themselves very well to excellent in reading in either language, their performance on the calculation of arithmetic task was not affected, rather was significant when the task was completed in the language the stimuli was presented. This analysis provided a greater understanding that reading the numeral value of the word in English and in Spanish did not play a role in this investigation on calculating of arithmetic sums.

As one can infer from Table 2 and/or Figure 1, when the vigilance task was subtracted from each arithmetic stimulus when presented in Arabic digit code the processing speed reached negative. Post vigilance time was necessary for the purpose of understanding on an average how long in milliseconds it can take an individual to calculate a certain task independent of codes. After observing such negative score on the

Arabic code, our group conducted a side project to observe what may be causing such differences, by conducting an experiment with 15 participants and randomizing the order in which to test the two vigilance task as well as the Arabic Exact and Arabic approximation task. In concluding the side project, we learned that when calculating digit in Arabic code, the processing speed was not significantly different from the vigilance task which only required participant to observe and respond by what symbols they have observed on the screen. Thus, calculating exact arithmetic of simple sums in Arabic code can be quite automatic which led to some scores being negative.

In order to observe language experience and performance on the arithmetic tasks, we analyzed the correlation between the reaction time of the arithmetic tasks and the participants self-rated proficiency in English and in Spanish. We expected correlation between self-rated language proficiency and the RTs from the verbal code in the exact arithmetic tasks. No associations between language proficiency and RTs should be expected in the performance of approximation tasks, supporting the triple-code model of exact arithmetic being language dependent in comparison to approximation which is language independent. Monolinguals in all cases have rated themselves 4 being quite well to 5 being excellent, while bilingual participants varied on how they rated themselves in each aspect of both languages. Neither group showed any significant correlation on how 32 they rated themselves and their processing speed for both task in Arabic code. While correlation does not show causation, referring to Table 3 we were able to observe a trend for bilingual participants on their self-rated proficiency and Spanish verbal tasks, in such that they negatively correlated with the processing speed in Spanish exact and how well they understand, speak, read, and write in Spanish. In addition, they negatively correlated with the processing speed in Spanish Approximation and how well they understand, read, and write in Spanish as it can be observed in Table 3. The negative correlation can be observed by a higher self-rated proficiency in Spanish and a lower reaction time to the arithmetic task in Spanish verbal code, for which bilinguals did not correlate with verbal code written in English. While for monolinguals negatively correlated with the processing speed in English Approximation and how well they speak, read, and write in

English as it can be observed in Table 4.

proficiency for for proficiency

language

.Correlations between the verbal and code the rated between self .Correlations

bilinguals Table

Table 4 . Correlations between the verbal code and self rated language proficiency for monolinguals

Monolingual Correlation to Arithmetic task and Proficiency English English English English English English Approx Exact

Understand Speak Read Write PV PV English 1 Understand English .768** 1 Speak English .714** .827** 1 Read English .681** .789** .737** 1 Write English -.063 -.228* -.319** -.212* 1 Approx PV English .002 -.031 -.173 -.097 .811** 1 Exact PV **. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed).

To better understand correlation of self reported language proficiency and the participants processing speed on the arithmetic code, a total proficiency score was calculated for each language. Total proficiency score for all participants in English was the sum of how well they read, write, speak and understand English, while for bilinguals, was calculated by the sum of how well they read, write, speak and understand Spanish.

As mentioned previously, each participant reported on a scale of 1 – 5 (1= virtually nothing to 5= excellent) on how well they read, write, speak and understand the language being tested. Correlation between total proficiency score and the arithmetic task for both groups are reported in Table 5 & 6, and is congruent to the correlation observed earlier between individual proficiency (read, write, speak and understand) and arithmetic codes.

A linear regression analysis was conducted to examine the extant of how much total

35 language proficiency score effectively predicted a processing speed on each arithmetic code and task. Total proficiency score for each language was used as the independent variable to observe influence on each of the arithmetic tasks on all codes. Total proficiency score was analyzed individually with each arithmetic code and task to observe its predictions on the processing speed. When total proficiency score in English language for monolinguals was analyzed as an independent factor with how they performed on English Exact and English approximation, the overall model explained a significant proportion of the variance (5.5%) in English Approximation (R2 = .055, F (1,

98) = 5.663, p < .05). Total proficiency score in English (b = -.234, t (98) = -2.38, p <

.05) emerges as a significant predictor of how well the monolingual participants performed in English Approximation, while not a significant predictor of performance in the English Exact arithmetic task. When total proficiency scores in English and in

Spanish language for bilinguals were analyzed as an independent factor with how they performed on English Exact, English approximation, Spanish Exact and Spanish

Approximation, the overall model explained a significant proportion of the variance

(13.8%) in Spanish Approximation (R2 = .138, F (1, 62) = 9.95, p < .001) and (11.5%) in

Spanish Exact (R2 = .115, F (1, 62) = 8.03, p < .001) when total proficiency score in

Spanish was the independent factor. Total proficiency score in Spanish (b = -.372, t (62)

= -3.15, p < .001) emerges as a significant predictor of how well the bilingual participants performed in Spanish Approximation and in Spanish Exact (b = -.339, t (62) = -2.83, p <

.01), while not being a significant predictor of performance in the English Exact and

English Approximation of arithmetic task. In addition no significant predictor can be reported at this time when total proficiency score in English language for bilinguals was

36 conducted against the dependent factor of how well they performed on the English verbal task.

Table 5. Correlations between the verbal code and the total self rated language proficiency for monolinguals

Total proficiency score and Verbal arithmetic task for monolinguals Total English English English Proficiency Approx PV Exact PV Total English 1 Proficiency English Approx PV -.234* 1 English Exact PV -.088 .811** 1 *. Correlation is significant at the 0.05 level (2-tailed). **. Correlation is significant at the 0.01 level (2-tailed).

Table 6 . Correlations between the verbal code and the total self rated language proficiency for bilinguals

Total proficiency score and arithmetic task for bilinguals Total Total English English Spanish English Spanish Approx Exact Approx Spanish Proficiency Proficiency PV PV PV Exact PV Total English 1 Proficiency Total Spanish -.087 1 Proficiency English -.176 -.089 1 Approx PV English Exact -.092 -.043 .753** 1 PV Spanish .170 -.372** .681** .576** 1 Approx PV Spanish Exact .247* -.339** .676** .697** .821** 1 PV *. Correlation is significant at the 0.05 level (2-tailed). **. Correlation is significant at the 0.01 level (2-tailed).

Table 7. Linear Regression for task and total proficiency

Group Task with Predictors B SE B β R2 F English Approximation PV Total English Proficiency - 57.47 24.15 - 0.23 0.06 5.66* Monolingual English Exact PV Total English Proficiency - 18.26 20.89 - 0.09 0.01 0.76 English Approximation PV Total English Proficiency - 32.54 23.11 - 0.18 0.03 1.98

English Exact PV Total English Proficiency - 19.85 27.25 - 0.09 0.01 0.53 Bilingual Spanish Approximation PV Total Spanish Proficiency - 72.16 22.88 - 0.37 0.14 9.95*

Spanish Exact PV Total Spanish Proficiency - 60.17 21.23 - 0.34 0.12 8.03* *. P< .05

DISCUSSION

The purpose of this investigation was to observe language experience and its influence on arithmetic processing using the triple-code model for arithmetic processing under previously tested Arabic digit code as well as newly created verbal code in English and in Spanish. It was hypothesized that since exact arithmetic is language dependent, participants would have a faster reaction time in comparison to approximation of arithmetic facts (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). Our investigation supports the hypothesis that exact arithmetic is language dependent, corroborating previous findings, as both groups (Monolinguals & Bilinguals) were observed to have a faster reaction time than compared to approximation of arithmetic facts independent of code.

While the performance during the Arabic digit code and the verbal code were significantly different, our investigation was unable to find a between group difference in which bilingual were significantly better than monolinguals. A possible explanation for not being able to observe between group differences may be due to the bilingual participants’ greater exposure to the English language than to Spanish, and/or being as proficient in the English language as the monolinguals. This explanation can be supported by the bilingual participant’s responses stating that they use the English language more often than Spanish in various situations as it can be observed in Table 8, as well as observing a slower reaction time when stimulus was presented in Spanish 39 compared to English (due to interference of both languages when observing the calculation task in Spanish). Previous research within bilingual populations were able to group the bilingual population by their proficiency in both languages in order to observe any differences that may exist when comparing to monolinguals (Kempert, Saalbach, &

Hardy, 2011). Thus, with a bilingual population with balanced proficiency in both and/or unbalanced proficiency in either languages (Higher proficiency in their first language than in their second language or opposite) may show differences in reaction time when compared to monolingual, which may be explored in future investigations.

In addition, from the scope of our investigation there was a within group difference in bilinguals in which they were faster in the arithmetic task written in English verbal code compared to Spanish. Bilingual groups faster reaction time in English verbal code compared to Spanish may be related to how often and in what condition both languages are used. Referring to Table 8 it can be deduced under various conditions that the bilingual participants used the English language more than Spanish, thus had a greater language exposure to English than Spanish. Greater language exposure to one language over another may have influenced a faster reaction time in English verbal task compared to the task written in Spanish verbal code.

In terms of correlation between self-rated proficiency in a language and the processing speed in the two tasks and/or code, we found trends that may suggest that being proficient in a language may aid the participant in the task when the stimulus is presented in the more proficient language of the participant. It has been theorized that participants exposure to two languages, when asked to perform a certain task in one of

40 the two languages, would have a first language advantage or perform the task very well if the task required the individual to complete a task in their more proficient language. This can be further proven with statements made by each participant after the completion of the entire paradigm to the experimenter. One participant mentioned that whether the arithmetic task was written in Arabic digit code or in any of the verbal code, they finished all tasks in Spanish, due to being exposed to Spanish more than English. In certain cases, bilingual participants asked to complete the English verbal tasks mentioned they translated the English verbal code in Spanish to better understand the task, prior to responding appropriately to the arithmetic sums. More often however, participants reported to the experimenter that when stimulus was presented in verbal code, they transcribed it into Arabic digits prior to responding with the appropriate sums. Further research should test how language switching due to greater language exposure can play a role on solving simple arithmetic sums when written in verbal code (Venkatraman, Siong,

Chee, & Ansari, 2006).

It can be deduced that language experience hindered performance on the arithmetic task for bilingual participants. In the questionnaire, each participant was asked which language they use to complete daily tasks listed in Table 8 and to what frequency.

Participants rated themselves (1= rarely, 2=sometimes, and 3=usually) on how often they watch television, read the newspaper, talk to friends, talk to relatives, and do mental calculation in one of their two spoken languages. It was observed that much of the tasks, besides talking to relatives, were done in English. Thus our bilingual population had greater English language exposure than Spanish, which may be responsible for the faster reaction time when doing arithmetic task in English than in Spanish. Such language 41 experience can also be a factor for much of the participants experiencing interference who might be exposed to both languages equally on a daily bases.

Table 8. Language in which daily task is completed by bilinguals participants

Daily Task Language task being Frequency completed Usually 52 (81.3 %) English Sometime 10 (15.6 %) Watch Television Rarely 2 (3.1 %) Usually 7 (10.9%) Spanish Sometime 27 (42.2 %) Rarely 25 (39.1 %) Usually 30 (46.9 %) English Sometime 10 (15.6%) Read Newspaper Rarely 20 (31.3 %) Usually 3 (4.7 %) Spanish Sometime 15 (23.4 %) Rarely 37 (57.8 %) Usually 57 (89.1 %) English Sometime 6 (9.4 %) Speak to friend Rarely 0 (0 %) Usually 25 (39.1 %) Spanish Sometime 18 (28.1 %) Rarely 17 (26.6 %) Usually 22 (34.4 %) English Sometime 22 (34.4 %) Speak to Relative Rarely 14 (21.9 %) Usually 45 (70.3 %) Spanish Sometime 12 (19.8 %) Rarely 6 (9.4 %) Usually 48 (75 %) English Sometime 10 (15.6) Mental Calculation Rarely 1 (1.6 %) Usually 8 (12.5 %) Spanish Sometime 14 (21.9 %) Rarely 38 (59.4 %)

With such evidence from our investigation, the triple-code model for arithmetic processing can be modified to take into account verbal code and how bilinguals interpret arithmetic facts and interference they may experience when stimulus is presented in one of the languages they speak. As mentioned earlier, the components of the direct route can be activated in a linguistically mediated operation such as visual identification (2 x 4), visuo-verbal transcoding (“two times four”), and verbal sequence completion (“two times four, eight”) (Dehaene, & Cohen, 1997). The model can be modified to take into account when bilinguals, who have access to two or more languages, may have interference between the languages during visuo-verbal transcoding of arithmetic operation written in verbal code. Such interference can lead the individual to experience first language advantage and can also be the time when the individual switches their language for a successful response to an arithmetic operation.

This investigation has supported the theory of double dissociation observed when processing arithmetic proposed by the triple-code model (Dehaene, & Cohen, 1997) by observing significant differences in processing speed when calculating arithmetic exact and approximation of simple sums even when the stimuli are presented in verbal code.

Also, we were able to find out through this investigation that some interference may exist with individuals who are bilingual when asked to perform simple arithmetic sums. For future studies, researchers can take into consideration the amount of time individuals are exposed to one language and any effect it has on the model, as participants in our investigation were not trained in any way such as reported by other researchers testing this model for arithmetic processing. From the scope of this study, we can infer that

43 language exposure or age of the participant may result in differences or hinder scores when performing the arithmetic task in verbal code.

There were several limitations to our investigation. The first was not having access to a population of Spanish speaking monolinguals to compare the scores of the monolingual English speaking population and the English-Spanish speaking bilingual population, in order to observe the differences in scores. In addition, year of language exposure may have had an effect on the scores due to the amount of years participants were exposed to numbers and their proficiency in Spanish. Using self-rated proficiency being subjective to one’s opinion verses using a quantitative measures to examine level of proficiency of the group in a particular language was another limitation. Greater language exposure, and/or experience in a language, may show a more significant difference between monolinguals and bilinguals if participants are tested for proficiency and are put into groups according to how proficient they are in their first language compared to their second language (High L1, Low L2), (High L1, High L2), (Low L1,

Low L2), and (Low L1, High L2) (Kempert, Saalbach, & Hardy, 2011).. Such limitations can be explored in future studies when observing language-switching while doing arithmetic problems that may be of interest individuals ability to interpret and manipulate arithmetic facts in a daily basis. Future implication of this research could aid organizations in interpreting various arithmetic tasks for monolinguals and bilinguals differently, thus giving the individual a fair chance at success in the task. In addition, after the analysis of this investigation, further research can investigate how an individual can interpret arithmetic facts when faced with word problems when various arithmetic algorithms need to be completed to have a successful phonological output of the task. 44

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