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For the article on Cognitive advantages of bilingualism:

Effects on L1 from prolonged exposure to L2
It has been suggested that prolonged naturalistic (DEFINE) exposure to L2 not only affects how a speaker's L2 is processed, but may also affect how their L1 is processed. For example, in immersion contexts (where the individual is in an environment where L2 is used), the individual experiences reduced access to L1 and extensive contact with L2, which (DELETED affects and) facilitates processing of L2. However, this may consequently affect processing of their L1 (e.g. increased difficulty in naming objects and phonology). Bilinguals are not two monolinguals in one brain; even the supposedly 'stable' L1 system is open to influence once individuals become proficient in the L2. If the level of immersion in L2 affects the language processing in L1, then bilinguals in the L1 environment should process sentences differently than similar bilinguals in the L2 environment. For example, a Spanish-dominant bilingual in Spain may process sentences differently than a Spanish-dominant bilingual in England (IS THIS EXAMPLE REALLY HELPFUL GIVEN THE FOLLOWING PARAGRAPH?).

To test this hypothesis, Dussias & Sagarra (2007) investigated how individuals in these two environments interpreted temporarily ambiguous phrases. For example, subjects were told "Alguien disparó al hijo de la actriz que estaba en el balcón!/ Someone shot the son of the actress who was on the balcony'. Subsequently, when asked the question "Quien estaba en el balcón?/ Who was on the balcony?," monolingual Spanish speakers will typically answer 'the son' as they have a high attachment preference (they attach the modifier to the 'higher' verb phrase [shot the son]). How do I make this more clear for non-linguists? Would syntax trees even make sense? Should I just say that since of the actress modifies son, it is lower? IDK? Whereas, monolingual English speakers will answer 'the actress' as they have a low attachment preference (they attach the modifier to the 'lower' verb phrase [the actress who was on the balcony]). Confirming the hypothesis, Duissas & Sagarra (2007) found that Spanish-English bilinguals in a Spanish-speaking environment showed preference for the typical Spanish high-attachment strategy and Spanish-English bilinguals in an English-speaking environment showed preference for the typical English low-attachment strategy, even when reading the phrase in Spanish (their dominant language). This may be because they have more exposure to English constructions, so this may be more available to them.

Studies relevant to Executive Function:

Building on Bialystok's research, Carlson and Meltzoff studied Spanish-English bilinguals. There were three groups: native bilinguals, English monolinguals, and English monolinguals enrolled in an immersion program. The bilingual children's scores were on par with the other groups, but when the two groups were adjusted for age, parent income and education, and verbal scores, the bilinguals outperformed the monolinguals on "conflict tasks," or tasks that required resolving multiple attention demands.

Bogulski, Rakoczy, Goodman, and Bialystok investigated how "lapsed bilinguals" (participants who used to be bilingual but are now monolingual) compared to monolinguals and fluent bilinguals in executive function tasks. The lapsed bilinguals tested better than monolinguals but worse than their fluent counterparts.

A study by Kalia, Wilbourn, and Ghio found that monolinguals and early bilinguals performed similarly on executive function tasks, while late bilinguals performed slightly worse.

Virginal Valian asserts that the correlations between bilingualism and executive function are inconsistent, largely due to the fact that different tasks contribute to executive function. Because some of these tasks are available to monolinguals and bilinguals may similarly participate in these tasks to varying degrees, she argues that bilinguals cannot be assumed superior to monolinguals in executive function. She also cites that bilinguals are not consistently better at all executive function tasks. Similarly, Paap and Greenberg assert that bilinguals are not necessariy superior at executive processing. They assess their sample as similar in confounding variables and found that not only was there not evidence supporting an advantage for bilinguals, but that if anything, the evidence would argue against this.

Perception
A review of the literature suggests that bilingualism has an additive effect on an individual's creativity, by enhancing their mental flexibility, their ability to solve problems, and to perceive situations in different ways and the ability to maintain or manipulate these perceptions to suit the task at hand, all in ways that matched monolingual peers do not exhibit.

One study addressed a less explored field of cognitive advantages bilingual children may exhibit, in the use of creativity to solve of mathematical problems. Participants were presented with problems that were either mathematical in nature (arranging two sets of bottle caps to be equal according to instruction) or non-mathematical (a common household problem represented in pictures) and were asked to provide solutions, while being rated on scales of creativity, flexibility and originality. The results of the study confirmed that the bilingual children were more creative in their problem solving than their monolingual peers. One attribution for this trait could be bilinguals' increased metalinguistic awareness, which creates a form of thinking that is more open and objective, resulting in increased awareness and flexibility.

This enhanced mental flexibility that develops in bilinguals influences more than their problem solving or linguistic skills. Language appears to change the way the world is perceived between individuals that speak different languages, and it has been shown to influence the perception of color as well as the categorisation of objects.

Thierry et al. studied how having different words for different colors in one language might affect the perception of that color as compared to a language that does not discriminate between those colors. In Greek, "light blue" is distinguished from "blue", not simply as a different shade but as a whole different category of color. In this study, bilingual and monolingual Greek/English participants were shown different shades of blue and light blue as well as green and light green (for which a distinction is not made in Greek) and ERPs were recorded. Electrophysiological measures showed a distinct pattern for the bilinguals indicating that they were perceiving the two colors as completely separate.

Cook et al. explored the fact that Japanese speakers are more likely than English ones to categorize objects according to their material as opposed to their shape. In their study, they found that the preferences of Japanese monolinguals learning English changed; the more proficient they became in English, the more their object categorization results matched those of English monolinguals.

Age of acquisition
A debate within the linguistic community is whether the age of acquiring one's L2 has effects on the cognitive advantages. A study on native bilingual vs late bilingual vs monolingual children in the USA revealed an overall bilingual advantage. Furthermore, native bilingual children demonstrated better performance on a selection of executive function tasks compared with their late bilingual and monolingual counterparts. Participants were controlled for age, verbal ability, and socioeconomic status (indicated by parent education level). However, there are various methodological outcomes which may question the validity of these results. Firstly, a small sample size was used, with only 12 children in the bilingual group, 21 in the late bilingual group, and 17 in the monolingual group. 'Late bilingual' in this study was classified as a monolingual child who had been in a bilingual school for 6 months (where half the lessons were in English and half in Spanish or Japanese). This may be a poor representation of 'late bilinguals,' as 6 months may not be enough time for cognitive changes and adaptations to the brain to have taken place, and these children will unlikely already be 'proficient' in the L2, therefore this may not an appropriate group sample to support the claims being made. In addition, the effect sizes on all the individual executive function tests were all small to moderate effect sizes (ƞ2= 0.01à 0.2). In combination with the lack of power due to small sample size, strong conclusions cannot be drawn from this data.

Another study, Kapa and Columbo (2013) investigated the attentional control of monolingual children, Spanish-English bilingual children who had learned both languages before the age of 3, and Spanish-English children who had learned English after age 3. Attentional control is a cognitive skill in which one can ignore unnecessary or impertinent information to the task at hand. Children were tested using an Attention Network Test. Although all groups obtained the same accuracy rates, the researchers found that early L2 learners (those who learned both languages before the age of 3) had the fastest reaction time. The late learners and monolinguals did not significantly differ on response time, illustrating that early L2 acquisition could be a decisive factor in executive control levels.