critical period hypothesis (cph)

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The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis

Jan vanhove.

Department of Multilingualism, University of Fribourg, Fribourg, Switzerland

Analyzed the data: JV. Wrote the paper: JV.

Associated Data

In second language acquisition research, the critical period hypothesis ( cph ) holds that the function between learners' age and their susceptibility to second language input is non-linear. This paper revisits the indistinctness found in the literature with regard to this hypothesis's scope and predictions. Even when its scope is clearly delineated and its predictions are spelt out, however, empirical studies–with few exceptions–use analytical (statistical) tools that are irrelevant with respect to the predictions made. This paper discusses statistical fallacies common in cph research and illustrates an alternative analytical method (piecewise regression) by means of a reanalysis of two datasets from a 2010 paper purporting to have found cross-linguistic evidence in favour of the cph . This reanalysis reveals that the specific age patterns predicted by the cph are not cross-linguistically robust. Applying the principle of parsimony, it is concluded that age patterns in second language acquisition are not governed by a critical period. To conclude, this paper highlights the role of confirmation bias in the scientific enterprise and appeals to second language acquisition researchers to reanalyse their old datasets using the methods discussed in this paper. The data and R commands that were used for the reanalysis are provided as supplementary materials.

Introduction

In the long term and in immersion contexts, second-language (L2) learners starting acquisition early in life – and staying exposed to input and thus learning over several years or decades – undisputedly tend to outperform later learners. Apart from being misinterpreted as an argument in favour of early foreign language instruction, which takes place in wholly different circumstances, this general age effect is also sometimes taken as evidence for a so-called ‘critical period’ ( cp ) for second-language acquisition ( sla ). Derived from biology, the cp concept was famously introduced into the field of language acquisition by Penfield and Roberts in 1959 [1] and was refined by Lenneberg eight years later [2] . Lenneberg argued that language acquisition needed to take place between age two and puberty – a period which he believed to coincide with the lateralisation process of the brain. (More recent neurological research suggests that different time frames exist for the lateralisation process of different language functions. Most, however, close before puberty [3] .) However, Lenneberg mostly drew on findings pertaining to first language development in deaf children, feral children or children with serious cognitive impairments in order to back up his claims. For him, the critical period concept was concerned with the implicit “automatic acquisition” [2, p. 176] in immersion contexts and does not preclude the possibility of learning a foreign language after puberty, albeit with much conscious effort and typically less success.

sla research adopted the critical period hypothesis ( cph ) and applied it to second and foreign language learning, resulting in a host of studies. In its most general version, the cph for sla states that the ‘susceptibility’ or ‘sensitivity’ to language input varies as a function of age, with adult L2 learners being less susceptible to input than child L2 learners. Importantly, the age–susceptibility function is hypothesised to be non-linear. Moving beyond this general version, we find that the cph is conceptualised in a multitude of ways [4] . This state of affairs requires scholars to make explicit their theoretical stance and assumptions [5] , but has the obvious downside that critical findings risk being mitigated as posing a problem to only one aspect of one particular conceptualisation of the cph , whereas other conceptualisations remain unscathed. This overall vagueness concerns two areas in particular, viz. the delineation of the cph 's scope and the formulation of testable predictions. Delineating the scope and formulating falsifiable predictions are, needless to say, fundamental stages in the scientific evaluation of any hypothesis or theory, but the lack of scholarly consensus on these points seems to be particularly pronounced in the case of the cph . This article therefore first presents a brief overview of differing views on these two stages. Then, once the scope of their cph version has been duly identified and empirical data have been collected using solid methods, it is essential that researchers analyse the data patterns soundly in order to assess the predictions made and that they draw justifiable conclusions from the results. As I will argue in great detail, however, the statistical analysis of data patterns as well as their interpretation in cph research – and this includes both critical and supportive studies and overviews – leaves a great deal to be desired. Reanalysing data from a recent cph -supportive study, I illustrate some common statistical fallacies in cph research and demonstrate how one particular cph prediction can be evaluated.

Delineating the scope of the critical period hypothesis

First, the age span for a putative critical period for language acquisition has been delimited in different ways in the literature [4] . Lenneberg's critical period stretched from two years of age to puberty (which he posits at about 14 years of age) [2] , whereas other scholars have drawn the cutoff point at 12, 15, 16 or 18 years of age [6] . Unlike Lenneberg, most researchers today do not define a starting age for the critical period for language learning. Some, however, consider the possibility of the critical period (or a critical period for a specific language area, e.g. phonology) ending much earlier than puberty (e.g. age 9 years [1] , or as early as 12 months in the case of phonology [7] ).

Second, some vagueness remains as to the setting that is relevant to the cph . Does the critical period constrain implicit learning processes only, i.e. only the untutored language acquisition in immersion contexts or does it also apply to (at least partly) instructed learning? Most researchers agree on the former [8] , but much research has included subjects who have had at least some instruction in the L2.

Third, there is no consensus on what the scope of the cp is as far as the areas of language that are concerned. Most researchers agree that a cp is most likely to constrain the acquisition of pronunciation and grammar and, consequently, these are the areas primarily looked into in studies on the cph [9] . Some researchers have also tried to define distinguishable cp s for the different language areas of phonetics, morphology and syntax and even for lexis (see [10] for an overview).

Fourth and last, research into the cph has focused on ‘ultimate attainment’ ( ua ) or the ‘final’ state of L2 proficiency rather than on the rate of learning. From research into the rate of acquisition (e.g. [11] – [13] ), it has become clear that the cph cannot hold for the rate variable. In fact, it has been observed that adult learners proceed faster than child learners at the beginning stages of L2 acquisition. Though theoretical reasons for excluding the rate can be posited (the initial faster rate of learning in adults may be the result of more conscious cognitive strategies rather than to less conscious implicit learning, for instance), rate of learning might from a different perspective also be considered an indicator of ‘susceptibility’ or ‘sensitivity’ to language input. Nevertheless, contemporary sla scholars generally seem to concur that ua and not rate of learning is the dependent variable of primary interest in cph research. These and further scope delineation problems relevant to cph research are discussed in more detail by, among others, Birdsong [9] , DeKeyser and Larson-Hall [14] , Long [10] and Muñoz and Singleton [6] .

Formulating testable hypotheses

Once the relevant cph 's scope has satisfactorily been identified, clear and testable predictions need to be drawn from it. At this stage, the lack of consensus on what the consequences or the actual observable outcome of a cp would have to look like becomes evident. As touched upon earlier, cph research is interested in the end state or ‘ultimate attainment’ ( ua ) in L2 acquisition because this “determines the upper limits of L2 attainment” [9, p. 10]. The range of possible ultimate attainment states thus helps researchers to explore the potential maximum outcome of L2 proficiency before and after the putative critical period.

One strong prediction made by some cph exponents holds that post- cp learners cannot reach native-like L2 competences. Identifying a single native-like post- cp L2 learner would then suffice to falsify all cph s making this prediction. Assessing this prediction is difficult, however, since it is not clear what exactly constitutes sufficient nativelikeness, as illustrated by the discussion on the actual nativelikeness of highly accomplished L2 speakers [15] , [16] . Indeed, there exists a real danger that, in a quest to vindicate the cph , scholars set the bar for L2 learners to match monolinguals increasingly higher – up to Swiftian extremes. Furthermore, the usefulness of comparing the linguistic performance in mono- and bilinguals has been called into question [6] , [17] , [18] . Put simply, the linguistic repertoires of mono- and bilinguals differ by definition and differences in the behavioural outcome will necessarily be found, if only one digs deep enough.

A second strong prediction made by cph proponents is that the function linking age of acquisition and ultimate attainment will not be linear throughout the whole lifespan. Before discussing how this function would have to look like in order for it to constitute cph -consistent evidence, I point out that the ultimate attainment variable can essentially be considered a cumulative measure dependent on the actual variable of interest in cph research, i.e. susceptibility to language input, as well as on such other factors like duration and intensity of learning (within and outside a putative cp ) and possibly a number of other influencing factors. To elaborate, the behavioural outcome, i.e. ultimate attainment, can be assumed to be integrative to the susceptibility function, as Newport [19] correctly points out. Other things being equal, ultimate attainment will therefore decrease as susceptibility decreases. However, decreasing ultimate attainment levels in and by themselves represent no compelling evidence in favour of a cph . The form of the integrative curve must therefore be predicted clearly from the susceptibility function. Additionally, the age of acquisition–ultimate attainment function can take just about any form when other things are not equal, e.g. duration of learning (Does learning last up until time of testing or only for a more or less constant number of years or is it dependent on age itself?) or intensity of learning (Do learners always learn at their maximum susceptibility level or does this intensity vary as a function of age, duration, present attainment and motivation?). The integral of the susceptibility function could therefore be of virtually unlimited complexity and its parameters could be adjusted to fit any age of acquisition–ultimate attainment pattern. It seems therefore astonishing that the distinction between level of sensitivity to language input and level of ultimate attainment is rarely made in the literature. Implicitly or explicitly [20] , the two are more or less equated and the same mathematical functions are expected to describe the two variables if observed across a range of starting ages of acquisition.

But even when the susceptibility and ultimate attainment variables are equated, there remains controversy as to what function linking age of onset of acquisition and ultimate attainment would actually constitute evidence for a critical period. Most scholars agree that not any kind of age effect constitutes such evidence. More specifically, the age of acquisition–ultimate attainment function would need to be different before and after the end of the cp [9] . According to Birdsong [9] , three basic possible patterns proposed in the literature meet this condition. These patterns are presented in Figure 1 . The first pattern describes a steep decline of the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function up to the end of the cp and a practically non-existent age effect thereafter. Pattern 2 is an “unconventional, although often implicitly invoked” [9, p. 17] notion of the cp function which contains a period of peak attainment (or performance at ceiling), i.e. performance does not vary as a function of age, which is often referred to as a ‘window of opportunity’. This time span is followed by an unbounded decline in ua depending on aoa . Pattern 3 includes characteristics of patterns 1 and 2. At the beginning of the aoa range, performance is at ceiling. The next segment is a downward slope in the age function which ends when performance reaches its floor. Birdsong points out that all of these patterns have been reported in the literature. On closer inspection, however, he concludes that the most convincing function describing these age effects is a simple linear one. Hakuta et al. [21] sketch further theoretically possible predictions of the cph in which the mean performance drops drastically and/or the slope of the aoa – ua proficiency function changes at a certain point.

The graphs are based on based on Figure 2 in [9] .

Although several patterns have been proposed in the literature, it bears pointing out that the most common explicit prediction corresponds to Birdsong's first pattern, as exemplified by the following crystal-clear statement by DeKeyser, one of the foremost cph proponents:

[A] strong negative correlation between age of acquisition and ultimate attainment throughout the lifespan (or even from birth through middle age), the only age effect documented in many earlier studies, is not evidence for a critical period…[T]he critical period concept implies a break in the AoA–proficiency function, i.e., an age (somewhat variable from individual to individual, of course, and therefore an age range in the aggregate) after which the decline of success rate in one or more areas of language is much less pronounced and/or clearly due to different reasons. [22, p. 445].

DeKeyser and before him among others Johnson and Newport [23] thus conceptualise only one possible pattern which would speak in favour of a critical period: a clear negative age effect before the end of the critical period and a much weaker (if any) negative correlation between age and ultimate attainment after it. This ‘flattened slope’ prediction has the virtue of being much more tangible than the ‘potential nativelikeness’ prediction: Testing it does not necessarily require comparing the L2-learners to a native control group and thus effectively comparing apples and oranges. Rather, L2-learners with different aoa s can be compared amongst themselves without the need to categorise them by means of a native-speaker yardstick, the validity of which is inevitably going to be controversial [15] . In what follows, I will concern myself solely with the ‘flattened slope’ prediction, arguing that, despite its clarity of formulation, cph research has generally used analytical methods that are irrelevant for the purposes of actually testing it.

Inferring non-linearities in critical period research: An overview

Group mean or proportion comparisons

[T]he main differences can be found between the native group and all other groups – including the earliest learner group – and between the adolescence group and all other groups. However, neither the difference between the two childhood groups nor the one between the two adulthood groups reached significance, which indicates that the major changes in eventual perceived nativelikeness of L2 learners can be associated with adolescence. [15, p. 270].

Similar group comparisons aimed at investigating the effect of aoa on ua have been carried out by both cph advocates and sceptics (among whom Bialystok and Miller [25, pp. 136–139], Birdsong and Molis [26, p. 240], Flege [27, pp. 120–121], Flege et al. [28, pp. 85–86], Johnson [29, p. 229], Johnson and Newport [23, p. 78], McDonald [30, pp. 408–410] and Patowski [31, pp. 456–458]). To be clear, not all of these authors drew direct conclusions about the aoa – ua function on the basis of these groups comparisons, but their group comparisons have been cited as indicative of a cph -consistent non-continuous age effect, as exemplified by the following quote by DeKeyser [22] :

Where group comparisons are made, younger learners always do significantly better than the older learners. The behavioral evidence, then, suggests a non-continuous age effect with a “bend” in the AoA–proficiency function somewhere between ages 12 and 16. [22, p. 448].

The first problem with group comparisons like these and drawing inferences on the basis thereof is that they require that a continuous variable, aoa , be split up into discrete bins. More often than not, the boundaries between these bins are drawn in an arbitrary fashion, but what is more troublesome is the loss of information and statistical power that such discretisation entails (see [32] for the extreme case of dichotomisation). If we want to find out more about the relationship between aoa and ua , why throw away most of the aoa information and effectively reduce the ua data to group means and the variance in those groups?

Comparison of correlation coefficients

Correlation-based inferences about slope discontinuities have similarly explicitly been made by cph advocates and skeptics alike, e.g. Bialystok and Miller [25, pp. 136 and 140], DeKeyser and colleagues [22] , [44] and Flege et al. [45, pp. 166 and 169]. Others did not explicitly infer the presence or absence of slope differences from the subset correlations they computed (among others Birdsong and Molis [26] , DeKeyser [8] , Flege et al. [28] and Johnson [29] ), but their studies nevertheless featured in overviews discussing discontinuities [14] , [22] . Indeed, the most recent overview draws a strong conclusion about the validity of the cph 's ‘flattened slope’ prediction on the basis of these subset correlations:

In those studies where the two groups are described separately, the correlation is much higher for the younger than for the older group, except in Birdsong and Molis (2001) [ =  [26] , JV], where there was a ceiling effect for the younger group. This global picture from more than a dozen studies provides support for the non-continuity of the decline in the AoA–proficiency function, which all researchers agree is a hallmark of a critical period phenomenon. [22, p. 448].

In Johnson and Newport's specific case [23] , their correlation-based inference that ua levels off after puberty happened to be largely correct: the gjt scores are more or less randomly distributed around a near-horizontal trend line [26] . Ultimately, however, it rests on the fallacy of confusing correlation coefficients with slopes, which seriously calls into question conclusions such as DeKeyser's (cf. the quote above).

It can then straightforwardly be deduced that, other things equal, the aoa – ua correlation in the older group decreases as the ua variance in the older group increases relative to the ua variance in the younger group (Eq. 3).

Lower correlation coefficients in older aoa groups may therefore be largely due to differences in ua variance, which have been reported in several studies [23] , [26] , [28] , [29] (see [46] for additional references). Greater variability in ua with increasing age is likely due to factors other than age proper [47] , such as the concomitant greater variability in exposure to literacy, degree of education, motivation and opportunity for language use, and by itself represents evidence neither in favour of nor against the cph .

Regression approaches

Having demonstrated that neither group mean or proportion comparisons nor correlation coefficient comparisons can directly address the ‘flattened slope’ prediction, I now turn to the studies in which regression models were computed with aoa as a predictor variable and ua as the outcome variable. Once again, this category of studies is not mutually exclusive with the two categories discussed above.

In a large-scale study using self-reports and approximate aoa s derived from a sample of the 1990 U.S. Census, Stevens found that the probability with which immigrants from various countries stated that they spoke English ‘very well’ decreased curvilinearly as a function of aoa [48] . She noted that this development is similar to the pattern found by Johnson and Newport [23] but that it contains no indication of an “abruptly defined ‘critical’ or sensitive period in L2 learning” [48, p. 569]. However, she modelled the self-ratings using an ordinal logistic regression model in which the aoa variable was logarithmically transformed. Technically, this is perfectly fine, but one should be careful not to read too much into the non-linear curves found. In logistic models, the outcome variable itself is modelled linearly as a function of the predictor variables and is expressed in log-odds. In order to compute the corresponding probabilities, these log-odds are transformed using the logistic function. Consequently, even if the model is specified linearly, the predicted probabilities will not lie on a perfectly straight line when plotted as a function of any one continuous predictor variable. Similarly, when the predictor variable is first logarithmically transformed and then used to linearly predict an outcome variable, the function linking the predicted outcome variables and the untransformed predictor variable is necessarily non-linear. Thus, non-linearities follow naturally from Stevens's model specifications. Moreover, cph -consistent discontinuities in the aoa – ua function cannot be found using her model specifications as they did not contain any parameters allowing for this.

Using data similar to Stevens's, Bialystok and Hakuta found that the link between the self-rated English competences of Chinese- and Spanish-speaking immigrants and their aoa could be described by a straight line [49] . In contrast to Stevens, Bialystok and Hakuta used a regression-based method allowing for changes in the function's slope, viz. locally weighted scatterplot smoothing ( lowess ). Informally, lowess is a non-parametrical method that relies on an algorithm that fits the dependent variable for small parts of the range of the independent variable whilst guaranteeing that the overall curve does not contain sudden jumps (for technical details, see [50] ). Hakuta et al. used an even larger sample from the same 1990 U.S. Census data on Chinese- and Spanish-speaking immigrants (2.3 million observations) [21] . Fitting lowess curves, no discontinuities in the aoa – ua slope could be detected. Moreover, the authors found that piecewise linear regression models, i.e. regression models containing a parameter that allows a sudden drop in the curve or a change of its slope, did not provide a better fit to the data than did an ordinary regression model without such a parameter.

To sum up, I have argued at length that regression approaches are superior to group mean and correlation coefficient comparisons for the purposes of testing the ‘flattened slope’ prediction. Acknowledging the reservations vis-à-vis self-estimated ua s, we still find that while the relationship between aoa and ua is not necessarily perfectly linear in the studies discussed, the data do not lend unequivocal support to this prediction. In the following section, I will reanalyse data from a recent empirical paper on the cph by DeKeyser et al. [44] . The first goal of this reanalysis is to further illustrate some of the statistical fallacies encountered in cph studies. Second, by making the computer code available I hope to demonstrate how the relevant regression models, viz. piecewise regression models, can be fitted and how the aoa representing the optimal breakpoint can be identified. Lastly, the findings of this reanalysis will contribute to our understanding of how aoa affects ua as measured using a gjt .

Summary of DeKeyser et al. (2010)

I chose to reanalyse a recent empirical paper on the cph by DeKeyser et al. [44] (henceforth DK et al.). This paper lends itself well to a reanalysis since it exhibits two highly commendable qualities: the authors spell out their hypotheses lucidly and provide detailed numerical and graphical data descriptions. Moreover, the paper's lead author is very clear on what constitutes a necessary condition for accepting the cph : a non-linearity in the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function, with ua declining less strongly as a function of aoa in older, post- cp arrivals compared to younger arrivals [14] , [22] . Lastly, it claims to have found cross-linguistic evidence from two parallel studies backing the cph and should therefore be an unsuspected source to cph proponents.

The authors set out to test the following hypotheses:

• Hypothesis 1: For both the L2 English and the L2 Hebrew group, the slope of the age of arrival–ultimate attainment function will not be linear throughout the lifespan, but will instead show a marked flattening between adolescence and adulthood.
• Hypothesis 2: The relationship between aptitude and ultimate attainment will differ markedly for the young and older arrivals, with significance only for the latter. (DK et al., p. 417)

Both hypotheses were purportedly confirmed, which in the authors' view provides evidence in favour of cph . The problem with this conclusion, however, is that it is based on a comparison of correlation coefficients. As I have argued above, correlation coefficients are not to be confused with regression coefficients and cannot be used to directly address research hypotheses concerning slopes, such as Hypothesis 1. In what follows, I will reanalyse the relationship between DK et al.'s aoa and gjt data in order to address Hypothesis 1. Additionally, I will lay bare a problem with the way in which Hypothesis 2 was addressed. The extracted data and the computer code used for the reanalysis are provided as supplementary materials, allowing anyone interested to scrutinise and easily reproduce my whole analysis and carry out their own computations (see ‘supporting information’).

Data extraction

In order to verify whether we did in fact extract the data points to a satisfactory degree of accuracy, I computed summary statistics for the extracted aoa and gjt data and checked these against the descriptive statistics provided by DK et al. (pp. 421 and 427). These summary statistics for the extracted data are presented in Table 1 . In addition, I computed the correlation coefficients for the aoa – gjt relationship for the whole aoa range and for aoa -defined subgroups and checked these coefficients against those reported by DK et al. (pp. 423 and 428). The correlation coefficients computed using the extracted data are presented in Table 2 . Both checks strongly suggest the extracted data to be virtually identical to the original data, and Dr DeKeyser confirmed this to be the case in response to an earlier draft of the present paper (personal communication, 6 May 2013).

Results and Discussion

Modelling the link between age of onset of acquisition and ultimate attainment.

I first replotted the aoa and gjt data we extracted from DK et al.'s scatterplots and added non-parametric scatterplot smoothers in order to investigate whether any changes in slope in the aoa – gjt function could be revealed, as per Hypothesis 1. Figures 3 and ​ and4 4 show this not to be the case. Indeed, simple linear regression models that model gjt as a function of aoa provide decent fits for both the North America and the Israel data, explaining 65% and 63% of the variance in gjt scores, respectively. The parameters of these models are given in Table 3 .

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 1.

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 5.

To ensure that both segments are joined at the breakpoint, the predictor variable is first centred at the breakpoint value, i.e. the breakpoint value is subtracted from the original predictor variable values. For a blow-by-blow account of how such models can be fitted in r , I refer to an example analysis by Baayen [55, pp. 214–222].

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

Solid: regression with breakpoint at aoa 16 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

Solid: regression with breakpoint at aoa 6 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

In sum, a regression model that allows for changes in the slope of the the aoa – gjt function to account for putative critical period effects provides a somewhat better fit to the North American data than does an everyday simple regression model. The improvement in model fit is marginal, however, and including a breakpoint does not result in any detectable improvement of model fit to the Israel data whatsoever. Breakpoint models therefore fail to provide solid cross-linguistic support in favour of critical period effects: across both data sets, gjt can satisfactorily be modelled as a linear function of aoa .

On partialling out ‘age at testing’

As I have argued above, correlation coefficients cannot be used to test hypotheses about slopes. When the correct procedure is carried out on DK et al.'s data, no cross-linguistically robust evidence for changes in the aoa – gjt function was found. In addition to comparing the zero-order correlations between aoa and gjt , however, DK et al. computed partial correlations in which the variance in aoa associated with the participants' age at testing ( aat ; a potentially confounding variable) was filtered out. They found that these partial correlations between aoa and gjt , which are given in Table 9 , differed between age groups in that they are stronger for younger than for older participants. This, DK et al. argue, constitutes additional evidence in favour of the cph . At this point, I can no longer provide my own analysis of DK et al.'s data seeing as the pertinent data points were not plotted. Nevertheless, the detailed descriptions by DK et al. strongly suggest that the use of these partial correlations is highly problematic. Most importantly, and to reiterate, correlations (whether zero-order or partial ones) are actually of no use when testing hypotheses concerning slopes. Still, one may wonder why the partial correlations differ across age groups. My surmise is that these differences are at least partly the by-product of an imbalance in the sampling procedure.

The upshot of this brief discussion is that the partial correlation differences reported by DK et al. are at least partly the result of an imbalance in the sampling procedure: aoa and aat were simply less intimately tied for the young arrivals in the North America study than for the older arrivals with L2 English or for all of the L2 Hebrew participants. In an ideal world, we would like to fix aat or ascertain that it at most only weakly correlates with aoa . This, however, would result in a strong correlation between aoa and another potential confound variable, length of residence in the L2 environment, bringing us back to square one. Allowing for only moderate correlations between aoa and aat might improve our predicament somewhat, but even in that case, we should tread lightly when making inferences on the basis of statistical control procedures [61] .

On estimating the role of aptitude

Having shown that Hypothesis 1 could not be confirmed, I now turn to Hypothesis 2, which predicts a differential role of aptitude for ua in sla in different aoa groups. More specifically, it states that the correlation between aptitude and gjt performance will be significant only for older arrivals. The correlation coefficients of the relationship between aptitude and gjt are presented in Table 10 .

The problem with both the wording of Hypothesis 2 and the way in which it is addressed is the following: it is assumed that a variable has a reliably different effect in different groups when the effect reaches significance in one group but not in the other. This logic is fairly widespread within several scientific disciplines (see e.g. [62] for a discussion). Nonetheless, it is demonstrably fallacious [63] . Here we will illustrate the fallacy for the specific case of comparing two correlation coefficients.

Apart from not being replicated in the North America study, does this difference actually show anything? I contend that it does not: what is of interest are not so much the correlation coefficients, but rather the interactions between aoa and aptitude in models predicting gjt . These interactions could be investigated by fitting a multiple regression model in which the postulated cp breakpoint governs the slope of both aoa and aptitude. If such a model provided a substantially better fit to the data than a model without a breakpoint for the aptitude slope and if the aptitude slope changes in the expected direction (i.e. a steeper slope for post- cp than for younger arrivals) for different L1–L2 pairings, only then would this particular prediction of the cph be borne out.

Using data extracted from a paper reporting on two recent studies that purport to provide evidence in favour of the cph and that, according to its authors, represent a major improvement over earlier studies (DK et al., p. 417), it was found that neither of its two hypotheses were actually confirmed when using the proper statistical tools. As a matter of fact, the gjt scores continue to decline at essentially the same rate even beyond the end of the putative critical period. According to the paper's lead author, such a finding represents a serious problem to his conceptualisation of the cph [14] ). Moreover, although modelling a breakpoint representing the end of a cp at aoa 16 may improve the statistical model slightly in study on learners of English in North America, the study on learners of Hebrew in Israel fails to confirm this finding. In fact, even if we were to accept the optimal breakpoint computed for the Israel study, it lies at aoa 6 and is associated with a different geometrical pattern.

Diverging age trends in parallel studies with participants with different L2s have similarly been reported by Birdsong and Molis [26] and are at odds with an L2-independent cph . One parsimonious explanation of such conflicting age trends may be that the overall, cross-linguistic age trend is in fact linear, but that fluctuations in the data (due to factors unaccounted for or randomness) may sometimes give rise to a ‘stretched L’-shaped pattern ( Figure 1, left panel ) and sometimes to a ‘stretched 7’-shaped pattern ( Figure 1 , middle panel; see also [66] for a similar comment).

Importantly, the criticism that DeKeyser and Larsson-Hall levy against two studies reporting findings similar to the present [48] , [49] , viz. that the data consisted of self-ratings of questionable validity [14] , does not apply to the present data set. In addition, DK et al. did not exclude any outliers from their analyses, so I assume that DeKeyser and Larsson-Hall's criticism [14] of Birdsong and Molis's study [26] , i.e. that the findings were due to the influence of outliers, is not applicable to the present data either. For good measure, however, I refitted the regression models with and without breakpoints after excluding one potentially problematic data point per model. The following data points had absolute standardised residuals larger than 2.5 in the original models without breakpoints as well as in those with breakpoints: the participant with aoa 17 and a gjt score of 125 in the North America study and the participant with aoa 12 and a gjt score of 117 in the Israel study. The resultant models were virtually identical to the original models (see Script S1 ). Furthermore, the aoa variable was sufficiently fine-grained and the aoa – gjt curve was not ‘presmoothed’ by the prior aggregation of gjt across parts of the aoa range (see [51] for such a criticism of another study). Lastly, seven of the nine “problems with supposed counter-evidence” to the cph discussed by Long [5] do not apply either, viz. (1) “[c]onfusion of rate and ultimate attainment”, (2) “[i]nappropriate choice of subjects”, (3) “[m]easurement of AO”, (4) “[l]eading instructions to raters”, (6) “[u]se of markedly non-native samples making near-native samples more likely to sound native to raters”, (7) “[u]nreliable or invalid measures”, and (8) “[i]nappropriate L1–L2 pairings”. Problem No. 5 (“Assessments based on limited samples and/or “language-like” behavior”) may be apropos given that only gjt data were used, leaving open the theoretical possibility that other measures might have yielded a different outcome. Finally, problem No. 9 (“Faulty interpretation of statistical patterns”) is, of course, precisely what I have turned the spotlights on.

Conclusions

The critical period hypothesis remains a hotly contested issue in the psycholinguistics of second-language acquisition. Discussions about the impact of empirical findings on the tenability of the cph generally revolve around the reliability of the data gathered (e.g. [5] , [14] , [22] , [52] , [67] , [68] ) and such methodological critiques are of course highly desirable. Furthermore, the debate often centres on the question of exactly what version of the cph is being vindicated or debunked. These versions differ mainly in terms of its scope, specifically with regard to the relevant age span, setting and language area, and the testable predictions they make. But even when the cph 's scope is clearly demarcated and its main prediction is spelt out lucidly, the issue remains to what extent the empirical findings can actually be marshalled in support of the relevant cph version. As I have shown in this paper, empirical data have often been taken to support cph versions predicting that the relationship between age of acquisition and ultimate attainment is not strictly linear, even though the statistical tools most commonly used (notably group mean and correlation coefficient comparisons) were, crudely put, irrelevant to this prediction. Methods that are arguably valid, e.g. piecewise regression and scatterplot smoothing, have been used in some studies [21] , [26] , [49] , but these studies have been criticised on other grounds. To my knowledge, such methods have never been used by scholars who explicitly subscribe to the cph .

I suspect that what may be going on is a form of ‘confirmation bias’ [69] , a cognitive bias at play in diverse branches of human knowledge seeking: Findings judged to be consistent with one's own hypothesis are hardly questioned, whereas findings inconsistent with one's own hypothesis are scrutinised much more strongly and criticised on all sorts of points [70] – [73] . My reanalysis of DK et al.'s recent paper may be a case in point. cph exponents used correlation coefficients to address their prediction about the slope of a function, as had been done in a host of earlier studies. Finding a result that squared with their expectations, they did not question the technical validity of their results, or at least they did not report this. (In fact, my reanalysis is actually a case in point in two respects: for an earlier draft of this paper, I had computed the optimal position of the breakpoints incorrectly, resulting in an insignificant improvement of model fit for the North American data rather than a borderline significant one. Finding a result that squared with my expectations, I did not question the technical validity of my results – until this error was kindly pointed out to me by Martijn Wieling (University of Tübingen).) That said, I am keen to point out that the statistical analyses in this particular paper, though suboptimal, are, as far as I could gather, reported correctly, i.e. the confirmation bias does not seem to have resulted in the blatant misreportings found elsewhere (see [74] for empirical evidence and discussion). An additional point to these authors' credit is that, apart from explicitly identifying their cph version's scope and making crystal-clear predictions, they present data descriptions that actually permit quantitative reassessments and have a history of doing so (e.g. the appendix in [8] ). This leads me to believe that they analysed their data all in good conscience and to hope that they, too, will conclude that their own data do not, in fact, support their hypothesis.

I end this paper on an upbeat note. Even though I have argued that the analytical tools employed in cph research generally leave much to be desired, the original data are, so I hope, still available. This provides researchers, cph supporters and sceptics alike, with an exciting opportunity to reanalyse their data sets using the tools outlined in the present paper and publish their findings at minimal cost of time and resources (for instance, as a comment to this paper). I would therefore encourage scholars to engage their old data sets and to communicate their analyses openly, e.g. by voluntarily publishing their data and computer code alongside their articles or comments. Ideally, cph supporters and sceptics would join forces to agree on a protocol for a high-powered study in order to provide a truly convincing answer to a core issue in sla .

Supporting Information

aoa and gjt data extracted from DeKeyser et al.'s North America study.

aoa and gjt data extracted from DeKeyser et al.'s Israel study.

Script with annotated R code used for the reanalysis. All add-on packages used can be installed from within R.

Acknowledgments

I would like to thank Irmtraud Kaiser (University of Fribourg) for helping me to get an overview of the literature on the critical period hypothesis in second language acquisition. Thanks are also due to Martijn Wieling (currently University of Tübingen) for pointing out an error in the R code accompanying an earlier draft of this paper.

Funding Statement

No current external funding sources for this study.

Multilingual Pedagogy and World Englishes

Linguistic Variety, Global Society

Critical Period Hypothesis (CPH)

Tom Scovel writes, “The CPH [critical period hypothesis] is conceivably the most contentious issue in SLA because there is disagreement over its exact age span; people disagree strenuously over which facets of language are affected; there are competing explanations for its existence; and, to top it off, many people don’t believe it exists at all” (113). Proposed by Wilder Penfield and Lamar Roberts in 1959, the Critical Period Hypothesis (CPH) argues that there is a specific period of time in which people can learn a language without traces of the L1 (a so-called “foreign” accent or even L1 syntactical features) manifesting in L2 production (Scovel 48). If a learner’s goal is to sound “native,” there may be age-related limitations or “maturational constraints” as Kenneth Hyltenstam and Niclas Abrahamsson call them, on how “native” they can sound. Reducing the impression left by the L1 is certainly possible after puberty, but eliminating that impression entirely may not be possible.

Kenji Hakuta et al. explains that the relationship between age and L1 interference in L2 production is really not up for debate:

“The diminished average achievement of older learners is supported by personal anecdote and documented by empirical evidence….What is controversial, though, is whether this pattern meets the conditions for concluding that a critical period constrains learning in a way predicted by the theory” (31).

Some learners manage to overcome the “constraints” that Scovel believes are “probably accounted for by neurological factors that are genetically specified in our species” (114), but these learners are exceptional rather than the rule. It may be biology; it may be due to something else. The debate will continue, but evidence seems to indicate that the older learners become, the more difficult complete acquisition can be.

“David Birdsong, Looking Inside and Beyond the Critical Period Hypothesis.”  YouTube,  uploaded by IWL Channel, 09 May 2016, https://www.youtube.com/watch?v=9Bo0C4dj7Mw.

Application

Instructors should consider taking the CPH into account when assessing their students’ oral communication in the target language. When “maturational constraints” are a potential concern, it seems more fair for instructors to weight comprehension more heavily than nativeness. A thorough understanding of the CPH can also help instructors to counteract adult learners’ “self-handicapping” by helping the learners understand that, in spite of constraints due to aging, they are still capable of acquiring many–if not most–aspects of the target language.

Bibliography

Hakuta, Kenji, et al. “Critical Evidence: A Test of the Critical-Period Hypothesis for Second-Language Acquisition.”  Psychological Science , vol. 14, no. 1, 2003, pp. 31–38.  JSTOR , www.jstor.org/stable/40063748.

Hyltenstam, Kenneth, and Niclas Abrahamsson. “Comments on Stefka H. Marinova-Todd, D. Bradford Marshall, and Catherine E. Snow’s ‘Three Misconceptions about Age and L2 Learning’: Age and L2 Learning: The Hazards of Matching Practical ‘Implications’ with Theoretical ‘Facts.’”  TESOL Quarterly , vol. 35, no. 1, 2001, pp. 151–170.  JSTOR , www.jstor.org/stable/3587863.

Nemer, Randa. “Critical Period Hypothesis.”  Prezi,  04 Dec. 2013, https://prezi.com/zzuch40ibrlq/critical-period-hypothesis-sla/#.

Scovel, Tom.  Learning New Languages . Heinle & Heinle, 2001.

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Age and the critical period hypothesis

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Christian Abello-Contesse, Age and the critical period hypothesis, ELT Journal , Volume 63, Issue 2, April 2009, Pages 170–172, https://doi.org/10.1093/elt/ccn072

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In the field of second language acquisition (SLA), how specific aspects of learning a non-native language (L2) may be affected by when the process begins is referred to as the ‘age factor’. Because of the way age intersects with a range of social, affective, educational, and experiential variables, clarifying its relationship with learning rate and/or success is a major challenge.

There is a popular belief that children as L2 learners are ‘superior’ to adults ( Scovel 2000 ), that is, the younger the learner, the quicker the learning process and the better the outcomes. Nevertheless, a closer examination of the ways in which age combines with other variables reveals a more complex picture, with both favourable and unfavourable age-related differences being associated with early- and late-starting L2 learners ( Johnstone 2002 ).

The ‘critical period hypothesis’ (CPH) is a particularly relevant case in point. This is the claim that there is, indeed, an optimal period for language acquisition, ending at puberty. However, in its original formulation ( Lenneberg 1967 ), evidence for its existence was based on the relearning of impaired L1 skills, rather than the learning of a second language under normal circumstances.

Furthermore, although the age factor is an uncontroversial research variable extending from birth to death ( Cook 1995 ), and the CPH is a narrowly focused proposal subject to recurrent debate, ironically, it is the latter that tends to dominate SLA discussions ( García Lecumberri and Gallardo 2003 ), resulting in a number of competing conceptualizations. Thus, in the current literature on the subject ( Bialystok 1997 ; Richards and Schmidt 2002 ; Abello-Contesse et al. 2006), references can be found to (i) multiple critical periods (each based on a specific language component, such as age six for L2 phonology), (ii) the non-existence of one or more critical periods for L2 versus L1 acquisition, (iii) a ‘sensitive’ yet not ‘critical’ period, and (iv) a gradual and continual decline from childhood to adulthood.

It therefore needs to be recognized that there is a marked contrast between the CPH as an issue of continuing dispute in SLA, on the one hand, and, on the other, the popular view that it is an invariable ‘law’, equally applicable to any L2 acquisition context or situation. In fact, research indicates that age effects of all kinds depend largely on the actual opportunities for learning which are available within overall contexts of L2 acquisition and particular learning situations, notably the extent to which initial exposure is substantial and sustained ( Lightbown 2000 ).

Thus, most classroom-based studies have shown not only a lack of direct correlation between an earlier start and more successful/rapid L2 development but also a strong tendency for older children and teenagers to be more efficient learners. For example, in research conducted in the context of conventional school programmes, Cenoz (2003) and Muñoz (2006) have shown that learners whose exposure to the L2 began at age 11 consistently displayed higher levels of proficiency than those for whom it began at 4 or 8. Furthermore, comparable limitations have been reported for young learners in school settings involving innovative, immersion-type programmes, where exposure to the target language is significantly increased through subject-matter teaching in the L2 ( Genesee 1992 ; Abello-Contesse 2006 ). In sum, as Harley and Wang (1997) have argued, more mature learners are usually capable of making faster initial progress in acquiring the grammatical and lexical components of an L2 due to their higher level of cognitive development and greater analytical abilities.

In terms of language pedagogy, it can therefore be concluded that (i) there is no single ‘magic’ age for L2 learning, (ii) both older and younger learners are able to achieve advanced levels of proficiency in an L2, and (iii) the general and specific characteristics of the learning environment are also likely to be variables of equal or greater importance.

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Original research article, critical period in second language acquisition: the age-attainment geometry.

• Teachers College, Columbia University, New York City, NY, United States

One of the most fascinating, consequential, and far-reaching debates that have occurred in second language acquisition research concerns the Critical Period Hypothesis [ 1 ]. Although the hypothesis is generally accepted for first language acquisition, it has been hotly debated on theoretical, methodological, and practical grounds for second language acquisition, fueling studies reporting contradictory findings and setting off competing explanations. The central questions are: Are the observed age effects in ultimate attainment confined to a bounded period, and if they are, are they biologically determined or maturationally constrained? In this article, we take a sui generis , interdisciplinary approach that leverages our understanding of second language acquisition and of physics laws of energy conservation and angular momentum conservation, mathematically deriving the age-attainment geometry. The theoretical lens, termed Energy Conservation Theory for Second Language Acquisition, provides a macroscopic perspective on the second language learning trajectory across the human lifespan.

Introduction

The Critical Period Hypothesis (CPH), as proposed by [ 1 ], that nativelike proficiency is only attainable within a finite period, extending from early infancy to puberty, has generally been accepted in language development research, but more so for first language acquisition (L1A) than for second language acquisition (L2A).

In the context of L2A, there are two parallel facts that appear to compound the difficulty of establishing the validity of CPH. One is that there is a stark difference in the level of ultimate attainment between child and adult learners. “Children eventually reach a more native-like level of proficiency than learners who start learning a second language as adults” ([ 2 ], p . 360). But this fact exists alongside another fact, namely, that there are vast differences in ultimate attainment among older learners. [ 3 ] observed:

Although few adults, if any, are completely successful, and many fail miserably, there are many who achieve very high levels of proficiency, given enough time, input, and effort, and given the right attitude, motivation, and learning environment. ( p . 13).

The dual facets of inter-learner differential success are at the nexus of second language acquisition research. As [ 4 ] once noted:

One of the enduring and fascinating problems confronting researchers of second language acquisition is whether adults can ever acquire native-like competence in a second language, or whether this is an accomplishment reserved for children who start learning at a relatively early age. As a secondary issue, there is the question of whether those rare cases of native-like success reported amongst adult learners are indeed what they seem, and if they are, how it is that such people can be successful when the vast majority are palpably not. ( p . 219).

The primary question Kellerman raised here is, in essence, a critical period (CP) question, concerning differential attainment between child and adult learners, and his secondary question relates to differential attainment among adult learners.

As of this writing, neither question has been settled. Instead, the two phenomena are often seen conflated in debates, including taking evidence for one as counter-evidence for the other (see, e.g., [ 5 ]). By and large, it would seem that the debate has come down to a matter of interpretation; the same facts are interpreted differently as evidence for or against CPH (see, e.g., [ 5 – 7 ]). This state of affairs, tinted with ideological differences over the role of nature and/or nature in language development, continues to put a tangible understanding of either phenomenon out of reach, let alone a coherent understanding of both phenomena. In order to break out of the rut of ‘he said, she said,” we need to engage in systems thinking.

Our research sought to juxtapose child and adult learners, as some researchers have, conceptually, attempted (see, e.g., [ 8 – 11 ]). Specifically, we built on and extended an interdisciplinary model of L2A, Energy Conservation Theory for L2A (ECT-L2A) [ 12 , 13 ], originally developed to account for differential attainment among adult learners, to child learners. In so doing, we sought to gain a coherent understanding of the dual facets of inter-learner differential success in L2A, in addition to mathematically obtaining the geometry of the age-attainment function, a core concern of the CPH/L2A debate.

In what follows, we first provide a quick overview of the CPH research in L2A. We then introduce ECT-L2A. Next, we extend ECT-L2A to the age issue, mathematically deriving the age-attainment function. After that, we discuss the resultant geometry and the fundamental nature of CPH/L2A, and, more broadly, L2 attainment across the human lifespan. We conclude by suggesting a number of avenues for furthering the research on CPH within the framework of ECT-L2A.

However, before we proceed, it is necessary to note two “boundary conditions” we have set for our work. First, the linguistic domain in which we theorize inter-learner differential attainment concerns only the grammatical/computational aspects of language, or what [ 14 , 15 ] calls basic language cognition, which concerns aspects of language where native speakers show little variance. As [ 2 ] has aptly pointed out, much of the confusion in the CPH-L2A debate is attributable to a lack of agreement on the scope of linguistic areas affected by CP. Second, we are only concerned with naturalistic acquisition (i.e., acquisition happens in an input-rich or immersion environment), not instructed learning (i.e., an input-poor environment). These two assumptions are often absent in CPH/L2A research, leading to the different circumstances under which researchers interpret the CP notion and empirical results (for discussion, see [ 7 ]).

The critical period hypothesis in L2A

To date, two questions have dominated the research and debate on CPH/L2A: What counts as evidence of a critical period? What accounts for the age-attainment difference between younger learners and older learners? More than 4 decades of research on CPH/L2A- from [ 16 ] to [ 17 ] to [ 18 ]—have, in the main, found an inverse correlation between the age of acquisition (AoA) and the level of grammatical attainment (see also [ 19 ], for a meta-analysis); “the age of acquisition is strongly negatively correlated with ultimate second language proficiency for grammar as well as for pronunciation” ([ 20 ], p . 88).

However, views are almost orthogonal over whether the observed inverse correlation can count as evidence of CPH or the observed difference is attributable to brain maturation (see, e.g., [ 5 , 7 , 21 – 35 ]).

For some researchers, true evidence or falsification of CPH for L2A must be tied to whether or not late learners can attain a native-like level of proficiency (e.g., [ 36 ]). Others contend that the nativelikeness threshold, in spite of it being “the most central aspect of the CPH” ([ 2 ], p . 362), is problematic, arguing that monolingual-like native attainment is simply impossible for L2 learners [ 37 , 38 ]. Echoing this view, [ 39 ] offered:

[Sequential] bilinguals are not “two monolinguals in one” in any social, psycholinguistic, or cognitive neurofunctional sense. From this perspective, it is of questionable methodological value to quantify bilinguals’ linguistic attainment as a proportion of monolinguals’ attainment, with those bilinguals reaching 100% levels of attainment considered nativelike. ( p . 121).

In the meantime, empirical research into adult learners have consistently produced evidence of selective nativelike attainment, that is, nativelikeness is attained vis-à-vis some aspects of the target language but not others. These studies employed a variety of methodologies, including cross-sectional studies and longitudinal case studies (see, e.g., [ 40 – 56 ]). Some researchers (e.g., [ 55 , 57 ]) take the selective nativelikeness as falsifying evidence of CPH/L2A; other researchers disagree (see, e.g., [ 36 ]).

Leaving aside the vexed issue of nativelikeness, 1 Birdsong [ 58 ], among others, postulated that CPH/L2A must ultimately pass geometric tests: if studies comparing younger learners and older learners yield the geometry of a “stretched Z” for the age-attainment function, that would prove the validity of CPH/L2A, or falsify it, if otherwise. The stretched Z or inverted S [ 20 ] references a bounded period in which the organism exhibits heightened neural plasticity and sensitivity to linguistic stimuli from the environment. This period has certain temporal and geometric features. Temporally, it extends from early infancy to puberty, coinciding with the time during which the brain undergoes maturation [ 1 , 36 , 59 – 62 ]. Geometrically, this period should exhibit two points of inflection or discontinuities, viz, “an abrupt onset or increase of sensitivity, a plateau of peak sensitivity, followed by a gradual offset or decline, with subsequent flattening of the degree of sensitivity” ([ 58 ], p . 111).

By the temporal and geometric hallmarks, few studies seem to have confirmed CPH/L2A, not even those that have allegedly found stark evidence. A case in point is the [ 17 ] study, which reported what appears to be clear-cut evidence of CPH/L2A: r = −.87, p <.01 for the early age of arrival (AoA) group and r = −.16, p >.05 for the late AoA group. As Johnson and Newport described it, “test performance was linearly related to [AoA] up to puberty; after puberty, performance was low but highly variable and unrelated to [AoA],” which supports “the conclusion that a critical period for language acquisition extends its effects to second language acquisition” ( p . 60). However, this claim has been contested.

Focusing on the geometry of the results, [ 58 ] pointed out that the random distribution of test scores within the late AoA group “does not license the conclusion that “through adulthood the function is low and flat” or the corresponding interpretation that “the shape of the function thus supports the claim that the effects of age of acquisition are effects of the maturational state of the learner” ([ 17 ], p . 79)” ( p . 117). Birdsong argued that if CPH holds for L2A, the performance scores of the late AoA group should be distributed horizontally in addition to showing marginal correlation with age. Accordingly, the random distribution of scores could only be taken as indicative of “a lack of systematic relationship between the performance and the AoA and not of a “levelling off of ultimate performance among those exposed to the language after puberty” ([ 17 ], p . 79)” ([ 58 ], p . 118).

Interpreting the same study, other researchers such as [ 20 ] did not set their sights as much on the random distribution of the performance scores among the late learners as on the discontinuity between the early AoA and late AoA groups, arguing that the qualitative difference is sufficient evidence of CPH/L2A.

If geometric satisfaction is one flash point in CPH/L2A research, explaining random distribution of performance scores or, essentially, differential attainment among late learners counts as another. Analyses of late learners’ ultimate attainment (e.g., [ 10 , 22 , 26 , 43 , 63 – 67 ]) have yielded a host of cognitive, socio-psychological, or experiential factors that can be associated with inter-learner differential attainment among late learners. The question, then, is whether or not these non-age factors confound, or even interact with, the age or maturational effect (see discussion in [ 2 , 68 – 72 ]. As Newport [ 7 ] aptly asked, “why cannot other variables interact with age effects?” ( p . 929).

These are undoubtedly complex questions for which sophisticated solutions are needed—beyond the methodological repairs many have thought are solely needed in advancing CPH/L2A research (see, e.g., [ 19 , 67 ]). In the remainder of this article, we take a different tack to the age issue, adopting a theoretical, hybrid approach, ECT-L2A [ 12 , 13 ], to mathematically derive the age-attainment function.

Energy-Conservation Theory for L2A

ECT-L2A is a theoretical model originally developed to account for the divergent states of ultimate attainment in adult L2A [ 12 , 13 ]. Drawing on the physics laws of energy conservation and angular momentum conservation, it theorizes the dynamic transformation and conservation of internal energies (i.e., from the learner) and external energies (i.e., from the environment) in rendering the learner’s ultimate attainment. This model, thus, takes into account nature and nurture factors, and specifically, uses five parameters - the linguistic environment or input, learner motivation, learner aptitude, distance between the L1 and the target language (TL) and the developing learner—and their interaction to account for levels of L2 ultimate attainment.

ECT-L2A draws a number of parallels between mechanical energies and human learning energies: kinetic energy for motivation and aptitude energy, potential energy for environmental energy, 2 and centrifugal energy for L1-TL deviation energy (for discussion, see [ 12 ]). These energies each perform a unique yet dynamic role. As the learner progresses in the developmental process, the energies shift in their dominance, while the total energy remains constant.

Mathematically, ECT-L2A reads as follows:

where ζ r denotes the learner’s motivational energy, r the learner’s position in the learning process relative to the TL, η the distance between L1 and TL, and ρ the input of TL. According to Eq. 1 , the total learning energy, ∈ , comes from the sum of motivation energy ζ r , aptitude (a constant) Λ , deviation energy η 2 r 2 , and environmental energy - ρ r .

The energy types included in Eq. 1 are embodiments of nature and nurture contributions. The potential energy or TL traction, - ρ r , represents the external or environmental energy, while the kinetic or motivational energy, ζ r , along with aptitude, Λ , and the centrifugal or deviation energy η 2 r 2 represent the internal energies.

Under the overarching condition of the total energy being the same or conserved throughout the learning process, ϵ = constant , each type of energy performs a different role, with one converting to another over time as the position of the learner changes in the developmental process.

For mathematical and conceptual convenience, (1) is rewritten into (2) which contains the effective potential energy, U eff (r) .

where U e f f r = η 2 r 2 − ρ r . In other words, the effective potential energy is the sum of deviation energy and the potential energy (see further breakdown in the next section).

The L2A energy system as depicted here is true of every learner, meaning that the total energy is constant for a single learner. But the total energy varies from learner to learner. Accordingly, different learners may reach different levels of ultimate attainment (i.e., closer or more distant from the TL), r 0 . This is illustrated in Figure 1 , where r 0 and r 0 ′ represent the ultimate attainments for learners with different amounts of total energy, ϵ >0 or ϵ <0.

FIGURE 1 . Inter-learner differential ultimate attainment as a function of different amounts of total energy: ϵ >0; ϵ < 0; ϵ = ϵ min [ 12 , 13 ].

Key to understanding Figure 1 is that it is the individual’s total energy that determines their level of attainment. Of the three scenarios on display here, ECT-L2A is only concerned with the case of ϵ ≥0, which represents the unbound process (r 0 , ∞), ignoring the bounded processes of ϵ < 0; ε = ϵmin.

The central thesis of ECT-L2A, as expressed in Eq. 1 , is that the moment a learner begins to receive substantive exposure to the TL, s/he enters a ‘gravitational’ field or a developmental ecosystem in which s/he is initially driven by kinetic or motivational energy, increasingly subject to the traction of the potential or environmental energy, but eventually stonewalled by the deviation energy or centrifugal barrier, resulting in an asymptotic endstate. This trajectory is further elaborated below.

The developmental trajectory depicted and forecast by ECT-L2A

The L2A trajectory begins with the learner at the outset of the learning process or at infinity (r = ∞). Initially, their progression toward the central source, i.e., the TL, is driven almost entirely by their motivation energy and aptitude, as expressed in Eq. 3 .

As learning proceeds, but with r still large (i.e., the learner still distant from the target) and the deviation energy much weaker than the environmental energy, η 2 r 2 ≪ ρ r (due to the second power of r ), the motivation energy rises as a result of its “interaction” with the environmental energy− ρ r , in which case the environmental energy transfers to the motivation energy. Mathematically, this is expressed in Eq. 4 .

As learning further progresses, the environmental energy - ρ r becomes dominant before yielding to the deviation energy η 2 r 2 . Eventually, the deviation energy overrides the environmental energy, as expressed in Eq. 1 , repeated below as Eq. 5 for ease of reference.

The deviation energy is so powerful that it draws the learner away from the target and their learning reaches an asymptote, where their motivation energy becomes minimal, ζ ( r 0 ) = 0, as expressed in Eq. 6 .

At this point, all other energies submit to the deviation energy, including the initial motivation energy ζ (∞) and some of the potential or environmental energy. Consequently, further exposure to TL input would not be of substantive help, meaning that it would not move the learner markedly closer to the target.

Figure 2 gives a geometric expression of the L1-TL deviation η, which is akin to the angular momentum of an object moving in a central force field [ 73 – 75 ]. The deviation from the TL, signifying the distance between the L1 and the TL, varies with different L1-TL pairings. For example, the distance index, according to the Automated Similarity Judgment Program Database [ 76 ], is 90.25 for Italian and English but 100.33 for Italian and Chinese.

FIGURE 2 . Geometric description of the deviation parameter η .

Figure 3 illustrates differential ultimate attainment (indicated by r 0 ) as a function of the deviation parameter η. As η increases, the level of attainment is lower or the attainment is further away from the target ( r = 0).

FIGURE 3 . Effective potentials U eff with different values of η [ 12 , 13 ].

For adult L2A, ECT-L2A predicts, inter alia , that high attainment is possible but full attainment is not. In other words, near-nativelike attainment is possible, but complete-nativelike attainment is not. ECT-L2A also predicts that while motivation and aptitude are part and parcel of the total energy of a given L2 learner, their role is largely confined to the earlier stage of development. Most of all, ECT-L2A predicts that the L1-L2 deviation is what keeps L2 attainment at asymptote.

For L2 younger learners, ECT-L2A also makes a number of predictions to which we now turn.

ECT-L2A vis-à-vis younger learners

As highlighted above, the deviation energy is what leads L2 attainment to an asymptote. It follows that as long as η (i.e., the L1-TL distance) is non-zero, the learner’s ultimate attainment, r 0 , will always eventuate in an asymptote. As shown in Figure 3 , the larger the deviation r 0 , the more distant the ultimate attainment r 0 is from the TL. Put differently, a larger η portends that learning would reach an asymptote earlier or that the ultimate attainment would be less native-like. But how does that work for child L2A?

On the ECT-L2A account, it is the low η value that determines child learners’ superior attainment. In child L2 learners, the deviation is low, because of the incipient or underdeveloped L1. However, as the L1 develops, the η value grows until it becomes a constant, presumably happening around puberty 3 , hence coinciding with the offset of the critical period [ 1 ]. As shown in Figures 1 , 3 , the smaller the deviation, η, the closer r 0 (i.e., the ultimate level of attainment) is to the TL or the higher the ultimate attainment.

From Eq. 6 the ultimate attainment of any L2 learner, irrespective of age, can be mathematically derived:

where ε = ϵ – Λ (i.e., total energy minus aptitude). r 0 here again denotes ultimate attainment. The upper panel in Figure 4 displays the geometry of ultimate attainment as a function of deviation, η.

FIGURE 4 . Double non-linearity of r 0 η [ (A) : first non-linearity] and η t [ (B) : second non-linearity] at early AoA.

For a given child learner, η is a constant, but different child learners can have a different η value, depending on their AoA . Herein lies a crucial difference from adult learning where η is a constant for all learners because of their uniform late AoA or age of acquisition and because their L1 has solidified. Adult learning starts at a time when the deviation between their L1 and the TL has become fixed, so to speak, as a result of having mastered their L1 (see the lower panel of Figure 4 ).

Further, for child L2 learners, η is simultaneously a function of their AoA, a proxy for time ( t ), and can therefore be expressed as η(t). This deviation function of time varies in the range of 0 ≤ η t ≤ η max . Accordingly; Eq. 7 can be mathematically rewritten into (8):

Assuming that as t grows or as AoA increases, η increases slowly and smoothly from 0 to η max until it solidifies into a constant, which marks the onset of adult learning, η( t ) can mathematically be expressed as (9).

where a is a constant. The geometry of the deviation function of time is illustrated in the lower panel of Figure 4 .

Figure 4 displays a double non-linearity characterizing L2 acquisition by young learners, with (A) showing the first order of non-linearity of r 0 η , that is, ultimate attainment as a function of deviation or the L1-TL distance (computed via Eq. 7 ), and with (B) displaying the second order of non-linearity, η (t), that is, η changing with t , age of acquisition (computed through Eq. 9 ).

Figure 5 illustrates ultimate attainment as a function of AoA, r 0 (t), and its derivative against t , d r 0 d t , which naturally yields three distinct periods: a critical period, t critical ; a post-critical period, t p-critical ; and an adult learning period, t adult . Within the critical period, t critical , r 0 ≅ 0 , meaning there is no real difference in attainment as age of acquisition increases. But within the post-critical period, t p-critical , r 0 changes dramatically, with d r 0 d t peaking and waning until it drops to the level approximating that of the adult period. Within the adult period, t adult , r 0 remains a constant, as attainment levels off.

FIGURE 5 . Ultimate attainment (the blue line) as a function of age of acquisition ( t ) and its derivatives giving three distinct periods (the orange line).

ECT-L2A, therefore, identifies three learning periods. First, there is a critical period, t critical , within which attainment is nativelike, r 0 ≅ 0. Notice that the blue line in Figure 5 is the lowest during the critical period, signifying that the attainment converges on the target, but it is the highest during the adult period, meaning that the attainment diverges greatly from the target. The offset of the critical period is smooth rather than abrupt, with the impact of deviation, η , slowly emerging at its offset. During this period, the L1 is surfacing, yet with negligible deviation from the TL and weak in strength.

Key to understanding this account of the critical period is the double non-linearity: first, ultimate attainment as a function of L1-TL deviation ( r 0 ( η ), see (A) in Figure 4 ); and second, L1-TL deviation as a function of AoA ( η ( t ); see (B) in Figure 4 ). Crucially, this double non-linearity extends a critical “point” into a critical “period” .

Second, there is a post-critical period, t p-critical , 0 < r 0 ≤ r 0 ( η max ), within which, with advancing AoA, the L1-L2 deviation grows larger and stronger, resulting in ultimate attainment that is increasingly lower (i.e., increasingly non-nativelike). The change rate of r 0 , its first derivative to time, d r 0 d t , is dramatic, waxing and waning. As such, the post-critical period is more complex and nuanced than the critical period. During the post-critical period, as the learner’s L1 becomes increasingly robust and developed, the deviation becomes larger, resulting in a level of attainment increasingly away from the target (i.e., increasingly non-nativelike).

Third, there is an adult learning period, t adult , η = η max ≅ constant, where, despite the continuously advancing AoA, the deviation reaches its maximum and remains a constant, as benchmarked in indexes of crosslinguistic distance (see, e.g., the Automated Similarity Judgment Program Database [ 76 ]). As a result, L2 ultimate attainment turns asymptotic (for discussion, see [ 12 , 13 ]).

The three periods mathematically produced by ECT-L2A coincide with the stretched “Z” slope that some researchers have argued (e.g., [ 17 , 58 , 59 ]) constitutes the most unambiguous evidence for CPH/L2A, and by extension, for a maturationally-based account of the generic success or lack thereof (i.e., nativelike or non-nativelike L2 proficiency) in early versus late starters. For better illustration of the stretched “Z,” we can convert Figure 5 into Figure 6 , using Eq. 10 .

where a t t stands for level of attainment. According to Eq. 10 , the smaller the r 0 is, the higher the attainment is.

FIGURE 6 . Level of attainment as a function of AoA.

In sum, ECT-L2A mathematically establishes the critical period geometry. That said, the geometry, as seen in Figure 6 , exhibits anything but abrupt inflections; the phase transitions are gradual and smooth. The adult period, for example, does not exhibit a complete “flattening” but markedly lower attainment with continuous decline (cf. [ 7 , 23 , 28 ]). 4

Explaining CPH/L2A

As is clear from the above, on the ECT-L2A account of the critical period, η (i.e., L1-TL deviation) is considered an inter-learner variable and, at once, a proxy for age of acquisition, t . More profoundly, however, ECT-L2A associates η with neural plasticity or sensitivity (cf [ 77 ]). The relationship between plasticity, p ( t ), and deviation function, η (t) , is expressed as (11):

Thus, the relationship between plasticity and the deviation function is one of inverse correlation. During the critical period, η = η min (i.e., minimal L1-TL deviation) and p = p max (i.e., maximal plasticity); conversely, during the adult learning period, η = η max (i.e., maximal L1-TL deviation) and p = p min (i.e., minimal plasticity). In short, an increased deviation, η (t) , corresponds to a decrease of plasticity, p (t) , and vice versa , as illustrated in Figure 7 .

FIGURE 7 . Plasticity as a function of age of acquisition.

Illustrated in Figure 7 is that neural plasticity, first proposed by [ 78 ] as the underlying cause of CP, is at its highest during the critical period and, as [ 79 ] put it, it “endures within the confines of its onset and offset” ( p . 182). But it begins to decline and drops to a low level during the post-critical period, and remains low through the adult learning period. 5 It would, therefore, seem reasonable to call the first period “critical” and the second period “sensitive.” It is worth mentioning in passing that the post-critical or sensitive period has thus far received scant empirical attention in CPH/L2A research.

Temporally, following the [ 59 ] conjecture, the critical period should last through early childhood from birth to age six, and the sensitive period should offset around puberty (see also [ 2 , 20 , 36 , 67 , 71 ]). Crucially, both periods are circumscribed, exhibiting discontinuities, with the critical period exhibiting maximal sensitivity, the sensitive period declining, though, for the most part, still far greater, sensitivity than the adult learning period. This view of a changing underlying mechanism across the three periods of AoA and attainment resonates with the Language as a Complex Adaptive System perspective (see, e.g., [ 80 ]). [ 81 ], for example, noted that “the processing mechanisms that underlie [language development] … are fundamentally non-linear. This means that development itself will frequently have phase-like characteristics, that there may be periods of extreme sensitivity to input (‘critical periods’)” ( p . 431).

ECT-L2A as a unifying model

ECT-L2A, by virtue of identifying the L1-TL deviation, η, as a lynchpin for age effects, provides an explanation for the differential ultimate attainment of early versus late starters. Essentially, in early AoA, η is a temporal and neuro-functional proxy tied respectively to a developing L1 and to a changing age and changing neuroplasticity. In contrast, in late AoA, η is a constant, due to the L1 being fully developed and the brain fully mature. This takes care of the first facet of inter-learner differential attainment. What about the second facet, viz., the inter-learner differential attainment among late learners?

ECT-L2A (as expressed in Eq. 1 ) is a model of an ecosystem where there is an interplay between learner-internal and environmental energies. In line with the general finding from L2 research that individual difference variables are largely responsible for inter-learner differential attainment of nativelike proficiency in adult learners (see, e.g., [ 27 , 35 , 77 , 82 , 83 ]), ECT-L2A specifically ties motivation and aptitude to kinetic energy, only to provide a more nuanced picture of the changing magnitude of individual difference variables.

Figure 8 illustrates the twin facets of inter-learner differential attainment. First, attainment varies as a function of AoA. Second, attainment varies within and across the three learning periods as a function of individual learners with different amounts of total energy, ϵ 1 < ϵ 2 < ϵ 3 . As shown, individual differences play out the least among learners of AoA falling within the critical period but the most within the adult learning period, consistent with the general findings from L2 research (see, e.g., [ 2 , 3 , 43 , 63 , 65 , 67 , 84 , 85 ]). During the post-critical or sensitive period, individual differences are initially non-apparent but become more pronounced with increasing AoA. 6

FIGURE 8 . Level of attainment for total energies ϵ 1 < ϵ 2 < ϵ 3 .

ECT-L2A thus offers a coherent explanation for variable attainment in late learners. First and foremost, it posits that individual learners’ total energy or “carrying capacity” [ 86 ] is different, which leads to different levels of attainment. Second, although the internal (motivation and aptitude) and external (environment) energies interact over time, ultimately it is the deviation energy η 2 r 2 that dominates and stalls the learner at asymptote (see Eq. 6 ). This account provides a much more nuanced perspective on the role of individual differences than has been given in the current L2A literature.

Extant empirical studies investigating individual difference variables through correlation analysis have mostly projected a static view of the role (some of) the variables play in L2A. In contrast, ECT-L2A gives a dynamic view and, more importantly, an interactive view. In the end, the individual difference variables are part of a larger ecosystem within which they do not act alone, but rather interact with other energies (i.e., potential energy and deviation energy), waxing and waning as a result of energy conservation.

In this article, we engaged with a central concern in the ongoing heated debate on CPH/L2A, that is, the geometry of age differences. Within the framework of ECT-L2A, an interdisciplinary model of L2 attainment, we mathematically derived the age-attainment function and established the presence of a critical period in L2A. Importantly, this period is part of a developmental trajectory that comprises three learning periods: a critical period, a post-critical or sensitive period, and an adult period.

ECT-L2A has thus far demonstrated a stunning internal consistency in that it mathematically identifies younger learners’ superior performance to adult learners’ as well as the differential attainment among adult learners.

ECT-L2A, while in broad agreement with an entrenchment-transfer account from L2A research that essentializes the role of the L1 in L2 attainment (see, e.g., [ 5 , 11 , 87 – 90 ]), provides a dynamic account of that role and its varying contributions to the different age-related learning periods. Furthermore, ECT-L2A offers an interactive account whereby the L1, as part of the deviation energy, interacts with other types of learner-internal and learner-external energies. Above all, ECT-L2A, by virtue of summoning internal and external energies, gives a coherent explanation for the twin facets of inter-learner differential success—as respectively manifested between younger and older learners and among older learners.

Validation of ECT-L2A is, however, required. Many questions warrant investigation. On this note, Johnson and Newort’s view [ 17 ], in particular, that the goal of any L2A theory should be to account for three sets of facts—a) gradual decline of performance, b) the age at which a decline in performance is detected, and c) the nature of adult performance—resonates with us. Although ECT-L2A shines a light on all three, further work is clearly needed. More specific to the focus of the present article, three sets of questions can be asked in relation to the three learning periods ECT-L2A has identified.

In the spirit of promoting collective intelligence, we present a subset of these questions below in the hope that they will spark interest among researchers across disciplines and inspire close-up investigations leveraging a variety of methodologies.

First, for the critical period:

1. When does the decline of learning begin?

2. How does it relate to the status of L1?

3. What is plasticity like in this period?

4. What does plasticity entail?

5. How is it related to a developing L1 and a developing L2?

Answers to these questions can, at least in part, be found in the various literatures across disciplines. But approaching these questions in relation to one another—as opposed to discretely—would likely yield a more systematic, holistic and coherent understanding. Or perhaps, in search of answers to any of these questions, one may realize that the existing understanding is way too shallow or inadequate. For instance, [ 18 ] cited “a lack of interference from a well-learned first language” as one of the possible causes of the age-attainment function in younger versus older learners. But what has not yet been established is the nature of the younger learners’ L1. What does “well-learned” mean? Is it established or is it still developing? At minimum, it cannot be a unitary phenomenon, given the age span of young learners.

Second, for the post-critical or sensitive period, ECT-L2A mathematically identifies two sub-periods. Thus, questions such as the following should be examined:

6. What prompts the initial dramatic decline of attainment?

7. How does each of the sub-periods relate to the status of L1?

8. How does the decline relate to changing plasticity?

9. How does it relate to grammatical performance?

Third, for the adult learning period, questions such as the following warrant close engagement:

10. How do learners with the same L1 background differ from each other in their L2 ultimate attainment?

11. How do learners with different L1 backgrounds differ from one another in their L2 ultimate attainment?

12. How is the trajectory of each type of energy, endogenous or exogenous, related to the level of attainment?

Investigating these questions, among others, will lead us to a better understanding not only of the critical period but also of L2 learning over the arc of human life.

The theoretical and practical importance of gaining a robust and comprehensive understanding of how age affects the L2 learning outcome calls for systematic investigations. To that end, ECT-L2A has offered a systems thinking perspective and framework.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Acknowledgments

We greatly appreciate the insightful and perceptive comments made by the reviewers on an earlier version of this article, and take sole responsibility for any error or omission.

Conflict of interest

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.

Publisher’s note

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1 Despite the centrality of “nativelikeness” to the Critical Period Hypothesis [ 1 ], studies in L2A have increasingly moved away from the use of the term in favor of “the level of ultimate attainment” [ 2 ].

2 The potential energy in ECT-L2A is akin to gravitational potential energy. As such it defines the central source field, serving as the primary energy that dynamically converts to other types of energy: kinetic energy and centrifugal energy. Similarly, the potential energy of L2A defines the field of learning. It stands for TL environment or input, serving as primary energy, dynamically converting to motivational and L1-TL deviation energies. An essential premise of ECT-L2A is the existence of potential energy. This premise is consistent with that underpinning L2A studies on CPH and ultimate attainment.

3 That is when the L1 becomes entrenched.

4 Looking back on the [ 17 ] study, [ 7 ], taking account of developments in the intervening 3 decades in understanding changes in the brain during adulthood, updated the earlier assertation about the stability of age effects in adulthood, noting that “it is more accurate to hypothesize that L2 proficiency SHOULD continue to decline during adulthood” and that “a critical or sensitive period for language acquisition is not absolute or sudden” ( p . 929, emphasis in original). She further argued that “[t]he lack of flattening of age function at adulthood in many studies does not mean that learning is not constrained by biologically based maturational changes” (ibid).

5 The plasticity never completely disappears, but rather becomes asymptotic.

6 Age and attainment function appears to follow a power law in that age effects are greatest during the critical period, less so during the post-critical or sensitive period, and weakest during the adult learning period (see Figure 8 ). Similarly, Figure 7 exhibits a power law relationship between age and plasticity: Plasticity is at its peak during the critical period, declines during the post-critical or sensitive period, and plateaus in the adult learning period.

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Keywords: ultimate attainment, critical period, second language acquisition, physics laws, energy conservation, angular momentum conservation, inter-learner differential attainment

Citation: Han Z and Bao G (2023) Critical period in second language acquisition: The age-attainment geometry. Front. Phys. 11:1142584. doi: 10.3389/fphy.2023.1142584

Received: 11 January 2023; Accepted: 02 March 2023; Published: 20 March 2023.

Reviewed by:

Copyright © 2023 Han and Bao. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: ZhaoHong Han, [email protected]

This article is part of the Research Topic

Social Physics and the Dynamics of Second Language Acquisition

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In This Article Expand or collapse the "in this article" section Critical Periods

Introduction, general overviews.

• Critical Periods in Language: Background Readings
• First-Language Acquisition (L1A)
• Age and Second-Language Acquisition (L2A)
• Controversy around the Critical Period Hypothesis for Second-Language Acquisition (CPH/L2A)
• Critical Period Geometry and Timing in L2A
• Brain-Based Studies of Critical Periods in L2A
• Bilingualism
• First-Language Attrition
• Sign Language
• Foreign Language Education
• Animal Models of Critical Periods
• Absolute Pitch

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Critical Periods by David P. Birdsong LAST REVIEWED: 09 December 2020 LAST MODIFIED: 12 January 2023 DOI: 10.1093/obo/9780199772810-0139

A critical period is a bounded maturational span during which experiential factors interact with biological mechanisms to determine neurocognitive and behavioral outcomes. In humans, the construct of critical period (CP) is commonly applied to first-language (L1) and second-language (L2) development. Some language researchers hold that during a CP, various mechanisms are at work that result in successful language acquisition and language processing. Outside of the period, other factors and mechanisms are involved, resulting in deficits in acquisition and processing. Many researchers believe that L1 development is constrained by maturationally based CPs. However, this notion is more controversial in L2 acquisition research, where the Critical Period Hypothesis for L2 acquisition (CPH/L2A) is debated on empirical, theoretical, and methodological grounds. Bilingualism researchers study the possibility that CPs may govern the likelihood and degree of loss (attrition) of the L1 among bilinguals as they age. Studies of CPs in L1 acquisition and L2 acquisition have been conducted with learners of spoken languages, signed languages, and artificial languages. CP research is considered in educational policy, particularly in the context of foreign language instruction. On a terminological note, a distinction is sometimes drawn between “critical” and “sensitive” periods, the latter term denoting receptivity of the organism to shaping by experience (or, in certain studies, suggesting relatively mild effects). Some researchers use these terms interchangeably, while others use one but not the other. Here, “critical period” will be used as a cover term unless specific reference is being made to sensitive period.

Lillard and Erisir 2011 describes juvenile CPs in language, imprinting, and vision. The article includes an informative table covering seven levels of neural changes in the brain in juveniles versus adults, with notes on the time course of changes and affected brain areas in animal and human models. The authors observe that changes in neural architecture triggered by early versus late experiences differ in degree more than type, and that the variety of triggering experiences is reduced with age. A second table summarizes neuroanatomical, electrophysiological, and neuroimaging techniques for observing specific types of neuroplasticity. Knudsen 2004 is exceptionally informative with respect to: prerequisites for CPs; the properties, mechanisms, and timing of plasticity; reopening of critical periods; the roles of presence and absence of relevant stimulation; and sensitive periods versus critical periods. Knudsen points out that complex behaviors (which include language use) may be regulated by multiple CPs. Takesian and Hensch 2013 emphasizes the individual-level plasticity of the timing of CP onset, peak, and offset, which may vary according to excitatory/inhibitory circuit balance that is sensitive to drugs, sleep, trauma, and genetic perturbation (see also Werker and Hensch 2015 [cited under First-Language Acquisition (L1A) ). Reh, et al. 2020 examines how plasticity is regulated at multiple timescales during development and provides examples from language processing, mental illness, and recovery from brain injury. Gabard-Durnam and McLaughlin 2020 outlines a set of current approaches to the study of sensitive periods in humans. These approaches include environmental manipulations (deprivation, enrichment, substitution), plasticity manipulations via pharmacological intervention, and computational modeling. Frankenhuis and Walasek 2020 develops an evolutionary model that accounts for sensitive periods that occur beyond the early stages of ontogeny.

Frankenhuis, Willem E., and Nicole Walasek. 2020. Modeling the evolution of sensitive periods . Developmental Cognitive Neuroscience 41:100715.

DOI: 10.1016/j.dcn.2019.100715

Sensitive periods in mid-ontogeny are favored by natural selection as a function of the reliability of relevant environmental cues.

Gabard-Durnam, Laurel, and Katie A. McLaughlin. 2020. Sensitive periods in human development: Charting a course for the future. Current Opinion in Behavioral Sciences 36:120–128.

DOI: 10.1016/j.cobeha.2020.09.003

Figures 1, 2 and 3 and their captions are particularly informative.

Knudsen, Eric I. 2004. Sensitive periods in the development of brain and behavior. Journal of Cognitive Neuroscience 16.8: 1412–1425.

DOI: 10.1162/0898929042304796

Focuses on the role of experience in modifying neural circuits during periods of plasticity, leading to connectivity patterns that become stable and less energy intensive, and making up what Knudsen calls the “stability landscape.”

Lillard, Angeline S., and Alev Erisir. 2011. Old dogs learning new tricks: Neuroplasticity beyond the juvenile period. Developmental Review 31:207–239.

DOI: 10.1016/j.dr.2011.07.008

A largely uncritical review and synthesis of well-known studies.

Reh, Rebecca K., Brian G. Dias, Charles A. Nelson III, et al. 2020. Critical period regulation across multiple timescales . Proceedings of the National Academy of Sciences of the United States of America 117.38: 23242–23251.

DOI: 10.1073/pnas.1820836117

A diverse group of specialists’ account of neurobiological CP mechanisms in animals and humans. Notes that “cortical plasticity is not only influenced by an animal’s life experiences but may also be modified by that of the parents. This occurs via parental behavior during the offspring’s early postnatal life, the in utero environment during gestation, or modification of the parental or fetal germ cells” (p. 23246).

Takesian, Anne E., and Takao K. Hensch. 2013. Balancing plasticity/stability across brain development. Progress in Brain Research 207:3–34.

DOI: 10.1016/B978-0-444-63327-9.00001-1

Illuminates the dynamic between the intrinsic plasticity of CP and the stabilization of neural networks, which limits maladaptive proliferation of circuit rewiring past the CP.

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The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis

* E-mail: [email protected]

Affiliation Department of Multilingualism, University of Fribourg, Fribourg, Switzerland

• Jan Vanhove

• Published: July 25, 2013
• https://doi.org/10.1371/journal.pone.0069172
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17 Jul 2014: The PLOS ONE Staff (2014) Correction: The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis. PLOS ONE 9(7): e102922. https://doi.org/10.1371/journal.pone.0102922 View correction

In second language acquisition research, the critical period hypothesis ( cph ) holds that the function between learners' age and their susceptibility to second language input is non-linear. This paper revisits the indistinctness found in the literature with regard to this hypothesis's scope and predictions. Even when its scope is clearly delineated and its predictions are spelt out, however, empirical studies–with few exceptions–use analytical (statistical) tools that are irrelevant with respect to the predictions made. This paper discusses statistical fallacies common in cph research and illustrates an alternative analytical method (piecewise regression) by means of a reanalysis of two datasets from a 2010 paper purporting to have found cross-linguistic evidence in favour of the cph . This reanalysis reveals that the specific age patterns predicted by the cph are not cross-linguistically robust. Applying the principle of parsimony, it is concluded that age patterns in second language acquisition are not governed by a critical period. To conclude, this paper highlights the role of confirmation bias in the scientific enterprise and appeals to second language acquisition researchers to reanalyse their old datasets using the methods discussed in this paper. The data and R commands that were used for the reanalysis are provided as supplementary materials.

Citation: Vanhove J (2013) The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis. PLoS ONE 8(7): e69172. https://doi.org/10.1371/journal.pone.0069172

Editor: Stephanie Ann White, UCLA, United States of America

Received: May 7, 2013; Accepted: June 7, 2013; Published: July 25, 2013

Copyright: © 2013 Jan Vanhove. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: No current external funding sources for this study.

Competing interests: The author has declared that no competing interests exist.

Introduction

In the long term and in immersion contexts, second-language (L2) learners starting acquisition early in life – and staying exposed to input and thus learning over several years or decades – undisputedly tend to outperform later learners. Apart from being misinterpreted as an argument in favour of early foreign language instruction, which takes place in wholly different circumstances, this general age effect is also sometimes taken as evidence for a so-called ‘critical period’ ( cp ) for second-language acquisition ( sla ). Derived from biology, the cp concept was famously introduced into the field of language acquisition by Penfield and Roberts in 1959 [1] and was refined by Lenneberg eight years later [2] . Lenneberg argued that language acquisition needed to take place between age two and puberty – a period which he believed to coincide with the lateralisation process of the brain. (More recent neurological research suggests that different time frames exist for the lateralisation process of different language functions. Most, however, close before puberty [3] .) However, Lenneberg mostly drew on findings pertaining to first language development in deaf children, feral children or children with serious cognitive impairments in order to back up his claims. For him, the critical period concept was concerned with the implicit “automatic acquisition” [2, p. 176] in immersion contexts and does not preclude the possibility of learning a foreign language after puberty, albeit with much conscious effort and typically less success.

sla research adopted the critical period hypothesis ( cph ) and applied it to second and foreign language learning, resulting in a host of studies. In its most general version, the cph for sla states that the ‘susceptibility’ or ‘sensitivity’ to language input varies as a function of age, with adult L2 learners being less susceptible to input than child L2 learners. Importantly, the age–susceptibility function is hypothesised to be non-linear. Moving beyond this general version, we find that the cph is conceptualised in a multitude of ways [4] . This state of affairs requires scholars to make explicit their theoretical stance and assumptions [5] , but has the obvious downside that critical findings risk being mitigated as posing a problem to only one aspect of one particular conceptualisation of the cph , whereas other conceptualisations remain unscathed. This overall vagueness concerns two areas in particular, viz. the delineation of the cph 's scope and the formulation of testable predictions. Delineating the scope and formulating falsifiable predictions are, needless to say, fundamental stages in the scientific evaluation of any hypothesis or theory, but the lack of scholarly consensus on these points seems to be particularly pronounced in the case of the cph . This article therefore first presents a brief overview of differing views on these two stages. Then, once the scope of their cph version has been duly identified and empirical data have been collected using solid methods, it is essential that researchers analyse the data patterns soundly in order to assess the predictions made and that they draw justifiable conclusions from the results. As I will argue in great detail, however, the statistical analysis of data patterns as well as their interpretation in cph research – and this includes both critical and supportive studies and overviews – leaves a great deal to be desired. Reanalysing data from a recent cph -supportive study, I illustrate some common statistical fallacies in cph research and demonstrate how one particular cph prediction can be evaluated.

Delineating the scope of the critical period hypothesis

First, the age span for a putative critical period for language acquisition has been delimited in different ways in the literature [4] . Lenneberg's critical period stretched from two years of age to puberty (which he posits at about 14 years of age) [2] , whereas other scholars have drawn the cutoff point at 12, 15, 16 or 18 years of age [6] . Unlike Lenneberg, most researchers today do not define a starting age for the critical period for language learning. Some, however, consider the possibility of the critical period (or a critical period for a specific language area, e.g. phonology) ending much earlier than puberty (e.g. age 9 years [1] , or as early as 12 months in the case of phonology [7] ).

Second, some vagueness remains as to the setting that is relevant to the cph . Does the critical period constrain implicit learning processes only, i.e. only the untutored language acquisition in immersion contexts or does it also apply to (at least partly) instructed learning? Most researchers agree on the former [8] , but much research has included subjects who have had at least some instruction in the L2.

Third, there is no consensus on what the scope of the cp is as far as the areas of language that are concerned. Most researchers agree that a cp is most likely to constrain the acquisition of pronunciation and grammar and, consequently, these are the areas primarily looked into in studies on the cph [9] . Some researchers have also tried to define distinguishable cp s for the different language areas of phonetics, morphology and syntax and even for lexis (see [10] for an overview).

Fourth and last, research into the cph has focused on ‘ultimate attainment’ ( ua ) or the ‘final’ state of L2 proficiency rather than on the rate of learning. From research into the rate of acquisition (e.g. [11] – [13] ), it has become clear that the cph cannot hold for the rate variable. In fact, it has been observed that adult learners proceed faster than child learners at the beginning stages of L2 acquisition. Though theoretical reasons for excluding the rate can be posited (the initial faster rate of learning in adults may be the result of more conscious cognitive strategies rather than to less conscious implicit learning, for instance), rate of learning might from a different perspective also be considered an indicator of ‘susceptibility’ or ‘sensitivity’ to language input. Nevertheless, contemporary sla scholars generally seem to concur that ua and not rate of learning is the dependent variable of primary interest in cph research. These and further scope delineation problems relevant to cph research are discussed in more detail by, among others, Birdsong [9] , DeKeyser and Larson-Hall [14] , Long [10] and Muñoz and Singleton [6] .

Formulating testable hypotheses

Once the relevant cph 's scope has satisfactorily been identified, clear and testable predictions need to be drawn from it. At this stage, the lack of consensus on what the consequences or the actual observable outcome of a cp would have to look like becomes evident. As touched upon earlier, cph research is interested in the end state or ‘ultimate attainment’ ( ua ) in L2 acquisition because this “determines the upper limits of L2 attainment” [9, p. 10]. The range of possible ultimate attainment states thus helps researchers to explore the potential maximum outcome of L2 proficiency before and after the putative critical period.

One strong prediction made by some cph exponents holds that post- cp learners cannot reach native-like L2 competences. Identifying a single native-like post- cp L2 learner would then suffice to falsify all cph s making this prediction. Assessing this prediction is difficult, however, since it is not clear what exactly constitutes sufficient nativelikeness, as illustrated by the discussion on the actual nativelikeness of highly accomplished L2 speakers [15] , [16] . Indeed, there exists a real danger that, in a quest to vindicate the cph , scholars set the bar for L2 learners to match monolinguals increasingly higher – up to Swiftian extremes. Furthermore, the usefulness of comparing the linguistic performance in mono- and bilinguals has been called into question [6] , [17] , [18] . Put simply, the linguistic repertoires of mono- and bilinguals differ by definition and differences in the behavioural outcome will necessarily be found, if only one digs deep enough.

A second strong prediction made by cph proponents is that the function linking age of acquisition and ultimate attainment will not be linear throughout the whole lifespan. Before discussing how this function would have to look like in order for it to constitute cph -consistent evidence, I point out that the ultimate attainment variable can essentially be considered a cumulative measure dependent on the actual variable of interest in cph research, i.e. susceptibility to language input, as well as on such other factors like duration and intensity of learning (within and outside a putative cp ) and possibly a number of other influencing factors. To elaborate, the behavioural outcome, i.e. ultimate attainment, can be assumed to be integrative to the susceptibility function, as Newport [19] correctly points out. Other things being equal, ultimate attainment will therefore decrease as susceptibility decreases. However, decreasing ultimate attainment levels in and by themselves represent no compelling evidence in favour of a cph . The form of the integrative curve must therefore be predicted clearly from the susceptibility function. Additionally, the age of acquisition–ultimate attainment function can take just about any form when other things are not equal, e.g. duration of learning (Does learning last up until time of testing or only for a more or less constant number of years or is it dependent on age itself?) or intensity of learning (Do learners always learn at their maximum susceptibility level or does this intensity vary as a function of age, duration, present attainment and motivation?). The integral of the susceptibility function could therefore be of virtually unlimited complexity and its parameters could be adjusted to fit any age of acquisition–ultimate attainment pattern. It seems therefore astonishing that the distinction between level of sensitivity to language input and level of ultimate attainment is rarely made in the literature. Implicitly or explicitly [20] , the two are more or less equated and the same mathematical functions are expected to describe the two variables if observed across a range of starting ages of acquisition.

But even when the susceptibility and ultimate attainment variables are equated, there remains controversy as to what function linking age of onset of acquisition and ultimate attainment would actually constitute evidence for a critical period. Most scholars agree that not any kind of age effect constitutes such evidence. More specifically, the age of acquisition–ultimate attainment function would need to be different before and after the end of the cp [9] . According to Birdsong [9] , three basic possible patterns proposed in the literature meet this condition. These patterns are presented in Figure 1 . The first pattern describes a steep decline of the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function up to the end of the cp and a practically non-existent age effect thereafter. Pattern 2 is an “unconventional, although often implicitly invoked” [9, p. 17] notion of the cp function which contains a period of peak attainment (or performance at ceiling), i.e. performance does not vary as a function of age, which is often referred to as a ‘window of opportunity’. This time span is followed by an unbounded decline in ua depending on aoa . Pattern 3 includes characteristics of patterns 1 and 2. At the beginning of the aoa range, performance is at ceiling. The next segment is a downward slope in the age function which ends when performance reaches its floor. Birdsong points out that all of these patterns have been reported in the literature. On closer inspection, however, he concludes that the most convincing function describing these age effects is a simple linear one. Hakuta et al. [21] sketch further theoretically possible predictions of the cph in which the mean performance drops drastically and/or the slope of the aoa – ua proficiency function changes at a certain point.

• PPT PowerPoint slide
• PNG larger image
• TIFF original image

The graphs are based on based on Figure 2 in [9] .

https://doi.org/10.1371/journal.pone.0069172.g001

Although several patterns have been proposed in the literature, it bears pointing out that the most common explicit prediction corresponds to Birdsong's first pattern, as exemplified by the following crystal-clear statement by DeKeyser, one of the foremost cph proponents:

[A] strong negative correlation between age of acquisition and ultimate attainment throughout the lifespan (or even from birth through middle age), the only age effect documented in many earlier studies, is not evidence for a critical period…[T]he critical period concept implies a break in the AoA–proficiency function, i.e., an age (somewhat variable from individual to individual, of course, and therefore an age range in the aggregate) after which the decline of success rate in one or more areas of language is much less pronounced and/or clearly due to different reasons. [22, p. 445].

DeKeyser and before him among others Johnson and Newport [23] thus conceptualise only one possible pattern which would speak in favour of a critical period: a clear negative age effect before the end of the critical period and a much weaker (if any) negative correlation between age and ultimate attainment after it. This ‘flattened slope’ prediction has the virtue of being much more tangible than the ‘potential nativelikeness’ prediction: Testing it does not necessarily require comparing the L2-learners to a native control group and thus effectively comparing apples and oranges. Rather, L2-learners with different aoa s can be compared amongst themselves without the need to categorise them by means of a native-speaker yardstick, the validity of which is inevitably going to be controversial [15] . In what follows, I will concern myself solely with the ‘flattened slope’ prediction, arguing that, despite its clarity of formulation, cph research has generally used analytical methods that are irrelevant for the purposes of actually testing it.

Inferring non-linearities in critical period research: An overview

Group mean or proportion comparisons.

[T]he main differences can be found between the native group and all other groups – including the earliest learner group – and between the adolescence group and all other groups. However, neither the difference between the two childhood groups nor the one between the two adulthood groups reached significance, which indicates that the major changes in eventual perceived nativelikeness of L2 learners can be associated with adolescence. [15, p. 270].

Similar group comparisons aimed at investigating the effect of aoa on ua have been carried out by both cph advocates and sceptics (among whom Bialystok and Miller [25, pp. 136–139], Birdsong and Molis [26, p. 240], Flege [27, pp. 120–121], Flege et al. [28, pp. 85–86], Johnson [29, p. 229], Johnson and Newport [23, p. 78], McDonald [30, pp. 408–410] and Patowski [31, pp. 456–458]). To be clear, not all of these authors drew direct conclusions about the aoa – ua function on the basis of these groups comparisons, but their group comparisons have been cited as indicative of a cph -consistent non-continuous age effect, as exemplified by the following quote by DeKeyser [22] :

Where group comparisons are made, younger learners always do significantly better than the older learners. The behavioral evidence, then, suggests a non-continuous age effect with a “bend” in the AoA–proficiency function somewhere between ages 12 and 16. [22, p. 448].

The first problem with group comparisons like these and drawing inferences on the basis thereof is that they require that a continuous variable, aoa , be split up into discrete bins. More often than not, the boundaries between these bins are drawn in an arbitrary fashion, but what is more troublesome is the loss of information and statistical power that such discretisation entails (see [32] for the extreme case of dichotomisation). If we want to find out more about the relationship between aoa and ua , why throw away most of the aoa information and effectively reduce the ua data to group means and the variance in those groups?

Comparison of correlation coefficients.

Correlation-based inferences about slope discontinuities have similarly explicitly been made by cph advocates and skeptics alike, e.g. Bialystok and Miller [25, pp. 136 and 140], DeKeyser and colleagues [22] , [44] and Flege et al. [45, pp. 166 and 169]. Others did not explicitly infer the presence or absence of slope differences from the subset correlations they computed (among others Birdsong and Molis [26] , DeKeyser [8] , Flege et al. [28] and Johnson [29] ), but their studies nevertheless featured in overviews discussing discontinuities [14] , [22] . Indeed, the most recent overview draws a strong conclusion about the validity of the cph 's ‘flattened slope’ prediction on the basis of these subset correlations:

In those studies where the two groups are described separately, the correlation is much higher for the younger than for the older group, except in Birdsong and Molis (2001) [ =  [26] , JV], where there was a ceiling effect for the younger group. This global picture from more than a dozen studies provides support for the non-continuity of the decline in the AoA–proficiency function, which all researchers agree is a hallmark of a critical period phenomenon. [22, p. 448].

In Johnson and Newport's specific case [23] , their correlation-based inference that ua levels off after puberty happened to be largely correct: the gjt scores are more or less randomly distributed around a near-horizontal trend line [26] . Ultimately, however, it rests on the fallacy of confusing correlation coefficients with slopes, which seriously calls into question conclusions such as DeKeyser's (cf. the quote above).

https://doi.org/10.1371/journal.pone.0069172.g002

Lower correlation coefficients in older aoa groups may therefore be largely due to differences in ua variance, which have been reported in several studies [23] , [26] , [28] , [29] (see [46] for additional references). Greater variability in ua with increasing age is likely due to factors other than age proper [47] , such as the concomitant greater variability in exposure to literacy, degree of education, motivation and opportunity for language use, and by itself represents evidence neither in favour of nor against the cph .

Regression approaches.

Having demonstrated that neither group mean or proportion comparisons nor correlation coefficient comparisons can directly address the ‘flattened slope’ prediction, I now turn to the studies in which regression models were computed with aoa as a predictor variable and ua as the outcome variable. Once again, this category of studies is not mutually exclusive with the two categories discussed above.

In a large-scale study using self-reports and approximate aoa s derived from a sample of the 1990 U.S. Census, Stevens found that the probability with which immigrants from various countries stated that they spoke English ‘very well’ decreased curvilinearly as a function of aoa [48] . She noted that this development is similar to the pattern found by Johnson and Newport [23] but that it contains no indication of an “abruptly defined ‘critical’ or sensitive period in L2 learning” [48, p. 569]. However, she modelled the self-ratings using an ordinal logistic regression model in which the aoa variable was logarithmically transformed. Technically, this is perfectly fine, but one should be careful not to read too much into the non-linear curves found. In logistic models, the outcome variable itself is modelled linearly as a function of the predictor variables and is expressed in log-odds. In order to compute the corresponding probabilities, these log-odds are transformed using the logistic function. Consequently, even if the model is specified linearly, the predicted probabilities will not lie on a perfectly straight line when plotted as a function of any one continuous predictor variable. Similarly, when the predictor variable is first logarithmically transformed and then used to linearly predict an outcome variable, the function linking the predicted outcome variables and the untransformed predictor variable is necessarily non-linear. Thus, non-linearities follow naturally from Stevens's model specifications. Moreover, cph -consistent discontinuities in the aoa – ua function cannot be found using her model specifications as they did not contain any parameters allowing for this.

Using data similar to Stevens's, Bialystok and Hakuta found that the link between the self-rated English competences of Chinese- and Spanish-speaking immigrants and their aoa could be described by a straight line [49] . In contrast to Stevens, Bialystok and Hakuta used a regression-based method allowing for changes in the function's slope, viz. locally weighted scatterplot smoothing ( lowess ). Informally, lowess is a non-parametrical method that relies on an algorithm that fits the dependent variable for small parts of the range of the independent variable whilst guaranteeing that the overall curve does not contain sudden jumps (for technical details, see [50] ). Hakuta et al. used an even larger sample from the same 1990 U.S. Census data on Chinese- and Spanish-speaking immigrants (2.3 million observations) [21] . Fitting lowess curves, no discontinuities in the aoa – ua slope could be detected. Moreover, the authors found that piecewise linear regression models, i.e. regression models containing a parameter that allows a sudden drop in the curve or a change of its slope, did not provide a better fit to the data than did an ordinary regression model without such a parameter.

To sum up, I have argued at length that regression approaches are superior to group mean and correlation coefficient comparisons for the purposes of testing the ‘flattened slope’ prediction. Acknowledging the reservations vis-à-vis self-estimated ua s, we still find that while the relationship between aoa and ua is not necessarily perfectly linear in the studies discussed, the data do not lend unequivocal support to this prediction. In the following section, I will reanalyse data from a recent empirical paper on the cph by DeKeyser et al. [44] . The first goal of this reanalysis is to further illustrate some of the statistical fallacies encountered in cph studies. Second, by making the computer code available I hope to demonstrate how the relevant regression models, viz. piecewise regression models, can be fitted and how the aoa representing the optimal breakpoint can be identified. Lastly, the findings of this reanalysis will contribute to our understanding of how aoa affects ua as measured using a gjt .

Summary of DeKeyser et al. (2010)

I chose to reanalyse a recent empirical paper on the cph by DeKeyser et al. [44] (henceforth DK et al.). This paper lends itself well to a reanalysis since it exhibits two highly commendable qualities: the authors spell out their hypotheses lucidly and provide detailed numerical and graphical data descriptions. Moreover, the paper's lead author is very clear on what constitutes a necessary condition for accepting the cph : a non-linearity in the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function, with ua declining less strongly as a function of aoa in older, post- cp arrivals compared to younger arrivals [14] , [22] . Lastly, it claims to have found cross-linguistic evidence from two parallel studies backing the cph and should therefore be an unsuspected source to cph proponents.

The authors set out to test the following hypotheses:

• Hypothesis 1: For both the L2 English and the L2 Hebrew group, the slope of the age of arrival–ultimate attainment function will not be linear throughout the lifespan, but will instead show a marked flattening between adolescence and adulthood.
• Hypothesis 2: The relationship between aptitude and ultimate attainment will differ markedly for the young and older arrivals, with significance only for the latter. (DK et al., p. 417)

Both hypotheses were purportedly confirmed, which in the authors' view provides evidence in favour of cph . The problem with this conclusion, however, is that it is based on a comparison of correlation coefficients. As I have argued above, correlation coefficients are not to be confused with regression coefficients and cannot be used to directly address research hypotheses concerning slopes, such as Hypothesis 1. In what follows, I will reanalyse the relationship between DK et al.'s aoa and gjt data in order to address Hypothesis 1. Additionally, I will lay bare a problem with the way in which Hypothesis 2 was addressed. The extracted data and the computer code used for the reanalysis are provided as supplementary materials, allowing anyone interested to scrutinise and easily reproduce my whole analysis and carry out their own computations (see ‘supporting information’).

Data extraction

In order to verify whether we did in fact extract the data points to a satisfactory degree of accuracy, I computed summary statistics for the extracted aoa and gjt data and checked these against the descriptive statistics provided by DK et al. (pp. 421 and 427). These summary statistics for the extracted data are presented in Table 1 . In addition, I computed the correlation coefficients for the aoa – gjt relationship for the whole aoa range and for aoa -defined subgroups and checked these coefficients against those reported by DK et al. (pp. 423 and 428). The correlation coefficients computed using the extracted data are presented in Table 2 . Both checks strongly suggest the extracted data to be virtually identical to the original data, and Dr DeKeyser confirmed this to be the case in response to an earlier draft of the present paper (personal communication, 6 May 2013).

https://doi.org/10.1371/journal.pone.0069172.t001

https://doi.org/10.1371/journal.pone.0069172.t002

Results and Discussion

Modelling the link between age of onset of acquisition and ultimate attainment.

I first replotted the aoa and gjt data we extracted from DK et al.'s scatterplots and added non-parametric scatterplot smoothers in order to investigate whether any changes in slope in the aoa – gjt function could be revealed, as per Hypothesis 1. Figures 3 and 4 show this not to be the case. Indeed, simple linear regression models that model gjt as a function of aoa provide decent fits for both the North America and the Israel data, explaining 65% and 63% of the variance in gjt scores, respectively. The parameters of these models are given in Table 3 .

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 1.

https://doi.org/10.1371/journal.pone.0069172.g003

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 5.

https://doi.org/10.1371/journal.pone.0069172.g004

https://doi.org/10.1371/journal.pone.0069172.t003

To ensure that both segments are joined at the breakpoint, the predictor variable is first centred at the breakpoint value, i.e. the breakpoint value is subtracted from the original predictor variable values. For a blow-by-blow account of how such models can be fitted in r , I refer to an example analysis by Baayen [55, pp. 214–222].

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g005

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g006

https://doi.org/10.1371/journal.pone.0069172.t004

https://doi.org/10.1371/journal.pone.0069172.g007

Solid: regression with breakpoint at aoa 16 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g008

Solid: regression with breakpoint at aoa 6 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g009

https://doi.org/10.1371/journal.pone.0069172.t005

https://doi.org/10.1371/journal.pone.0069172.t006

https://doi.org/10.1371/journal.pone.0069172.t007

https://doi.org/10.1371/journal.pone.0069172.t008

In sum, a regression model that allows for changes in the slope of the the aoa – gjt function to account for putative critical period effects provides a somewhat better fit to the North American data than does an everyday simple regression model. The improvement in model fit is marginal, however, and including a breakpoint does not result in any detectable improvement of model fit to the Israel data whatsoever. Breakpoint models therefore fail to provide solid cross-linguistic support in favour of critical period effects: across both data sets, gjt can satisfactorily be modelled as a linear function of aoa .

On partialling out ‘age at testing’

As I have argued above, correlation coefficients cannot be used to test hypotheses about slopes. When the correct procedure is carried out on DK et al.'s data, no cross-linguistically robust evidence for changes in the aoa – gjt function was found. In addition to comparing the zero-order correlations between aoa and gjt , however, DK et al. computed partial correlations in which the variance in aoa associated with the participants' age at testing ( aat ; a potentially confounding variable) was filtered out. They found that these partial correlations between aoa and gjt , which are given in Table 9 , differed between age groups in that they are stronger for younger than for older participants. This, DK et al. argue, constitutes additional evidence in favour of the cph . At this point, I can no longer provide my own analysis of DK et al.'s data seeing as the pertinent data points were not plotted. Nevertheless, the detailed descriptions by DK et al. strongly suggest that the use of these partial correlations is highly problematic. Most importantly, and to reiterate, correlations (whether zero-order or partial ones) are actually of no use when testing hypotheses concerning slopes. Still, one may wonder why the partial correlations differ across age groups. My surmise is that these differences are at least partly the by-product of an imbalance in the sampling procedure.

https://doi.org/10.1371/journal.pone.0069172.t009

The upshot of this brief discussion is that the partial correlation differences reported by DK et al. are at least partly the result of an imbalance in the sampling procedure: aoa and aat were simply less intimately tied for the young arrivals in the North America study than for the older arrivals with L2 English or for all of the L2 Hebrew participants. In an ideal world, we would like to fix aat or ascertain that it at most only weakly correlates with aoa . This, however, would result in a strong correlation between aoa and another potential confound variable, length of residence in the L2 environment, bringing us back to square one. Allowing for only moderate correlations between aoa and aat might improve our predicament somewhat, but even in that case, we should tread lightly when making inferences on the basis of statistical control procedures [61] .

On estimating the role of aptitude

Having shown that Hypothesis 1 could not be confirmed, I now turn to Hypothesis 2, which predicts a differential role of aptitude for ua in sla in different aoa groups. More specifically, it states that the correlation between aptitude and gjt performance will be significant only for older arrivals. The correlation coefficients of the relationship between aptitude and gjt are presented in Table 10 .

https://doi.org/10.1371/journal.pone.0069172.t010

The problem with both the wording of Hypothesis 2 and the way in which it is addressed is the following: it is assumed that a variable has a reliably different effect in different groups when the effect reaches significance in one group but not in the other. This logic is fairly widespread within several scientific disciplines (see e.g. [62] for a discussion). Nonetheless, it is demonstrably fallacious [63] . Here we will illustrate the fallacy for the specific case of comparing two correlation coefficients.

Apart from not being replicated in the North America study, does this difference actually show anything? I contend that it does not: what is of interest are not so much the correlation coefficients, but rather the interactions between aoa and aptitude in models predicting gjt . These interactions could be investigated by fitting a multiple regression model in which the postulated cp breakpoint governs the slope of both aoa and aptitude. If such a model provided a substantially better fit to the data than a model without a breakpoint for the aptitude slope and if the aptitude slope changes in the expected direction (i.e. a steeper slope for post- cp than for younger arrivals) for different L1–L2 pairings, only then would this particular prediction of the cph be borne out.

Using data extracted from a paper reporting on two recent studies that purport to provide evidence in favour of the cph and that, according to its authors, represent a major improvement over earlier studies (DK et al., p. 417), it was found that neither of its two hypotheses were actually confirmed when using the proper statistical tools. As a matter of fact, the gjt scores continue to decline at essentially the same rate even beyond the end of the putative critical period. According to the paper's lead author, such a finding represents a serious problem to his conceptualisation of the cph [14] ). Moreover, although modelling a breakpoint representing the end of a cp at aoa 16 may improve the statistical model slightly in study on learners of English in North America, the study on learners of Hebrew in Israel fails to confirm this finding. In fact, even if we were to accept the optimal breakpoint computed for the Israel study, it lies at aoa 6 and is associated with a different geometrical pattern.

Diverging age trends in parallel studies with participants with different L2s have similarly been reported by Birdsong and Molis [26] and are at odds with an L2-independent cph . One parsimonious explanation of such conflicting age trends may be that the overall, cross-linguistic age trend is in fact linear, but that fluctuations in the data (due to factors unaccounted for or randomness) may sometimes give rise to a ‘stretched L’-shaped pattern ( Figure 1, left panel ) and sometimes to a ‘stretched 7’-shaped pattern ( Figure 1 , middle panel; see also [66] for a similar comment).

Importantly, the criticism that DeKeyser and Larsson-Hall levy against two studies reporting findings similar to the present [48] , [49] , viz. that the data consisted of self-ratings of questionable validity [14] , does not apply to the present data set. In addition, DK et al. did not exclude any outliers from their analyses, so I assume that DeKeyser and Larsson-Hall's criticism [14] of Birdsong and Molis's study [26] , i.e. that the findings were due to the influence of outliers, is not applicable to the present data either. For good measure, however, I refitted the regression models with and without breakpoints after excluding one potentially problematic data point per model. The following data points had absolute standardised residuals larger than 2.5 in the original models without breakpoints as well as in those with breakpoints: the participant with aoa 17 and a gjt score of 125 in the North America study and the participant with aoa 12 and a gjt score of 117 in the Israel study. The resultant models were virtually identical to the original models (see Script S1 ). Furthermore, the aoa variable was sufficiently fine-grained and the aoa – gjt curve was not ‘presmoothed’ by the prior aggregation of gjt across parts of the aoa range (see [51] for such a criticism of another study). Lastly, seven of the nine “problems with supposed counter-evidence” to the cph discussed by Long [5] do not apply either, viz. (1) “[c]onfusion of rate and ultimate attainment”, (2) “[i]nappropriate choice of subjects”, (3) “[m]easurement of AO”, (4) “[l]eading instructions to raters”, (6) “[u]se of markedly non-native samples making near-native samples more likely to sound native to raters”, (7) “[u]nreliable or invalid measures”, and (8) “[i]nappropriate L1–L2 pairings”. Problem No. 5 (“Assessments based on limited samples and/or “language-like” behavior”) may be apropos given that only gjt data were used, leaving open the theoretical possibility that other measures might have yielded a different outcome. Finally, problem No. 9 (“Faulty interpretation of statistical patterns”) is, of course, precisely what I have turned the spotlights on.

Conclusions

The critical period hypothesis remains a hotly contested issue in the psycholinguistics of second-language acquisition. Discussions about the impact of empirical findings on the tenability of the cph generally revolve around the reliability of the data gathered (e.g. [5] , [14] , [22] , [52] , [67] , [68] ) and such methodological critiques are of course highly desirable. Furthermore, the debate often centres on the question of exactly what version of the cph is being vindicated or debunked. These versions differ mainly in terms of its scope, specifically with regard to the relevant age span, setting and language area, and the testable predictions they make. But even when the cph 's scope is clearly demarcated and its main prediction is spelt out lucidly, the issue remains to what extent the empirical findings can actually be marshalled in support of the relevant cph version. As I have shown in this paper, empirical data have often been taken to support cph versions predicting that the relationship between age of acquisition and ultimate attainment is not strictly linear, even though the statistical tools most commonly used (notably group mean and correlation coefficient comparisons) were, crudely put, irrelevant to this prediction. Methods that are arguably valid, e.g. piecewise regression and scatterplot smoothing, have been used in some studies [21] , [26] , [49] , but these studies have been criticised on other grounds. To my knowledge, such methods have never been used by scholars who explicitly subscribe to the cph .

I suspect that what may be going on is a form of ‘confirmation bias’ [69] , a cognitive bias at play in diverse branches of human knowledge seeking: Findings judged to be consistent with one's own hypothesis are hardly questioned, whereas findings inconsistent with one's own hypothesis are scrutinised much more strongly and criticised on all sorts of points [70] – [73] . My reanalysis of DK et al.'s recent paper may be a case in point. cph exponents used correlation coefficients to address their prediction about the slope of a function, as had been done in a host of earlier studies. Finding a result that squared with their expectations, they did not question the technical validity of their results, or at least they did not report this. (In fact, my reanalysis is actually a case in point in two respects: for an earlier draft of this paper, I had computed the optimal position of the breakpoints incorrectly, resulting in an insignificant improvement of model fit for the North American data rather than a borderline significant one. Finding a result that squared with my expectations, I did not question the technical validity of my results – until this error was kindly pointed out to me by Martijn Wieling (University of Tübingen).) That said, I am keen to point out that the statistical analyses in this particular paper, though suboptimal, are, as far as I could gather, reported correctly, i.e. the confirmation bias does not seem to have resulted in the blatant misreportings found elsewhere (see [74] for empirical evidence and discussion). An additional point to these authors' credit is that, apart from explicitly identifying their cph version's scope and making crystal-clear predictions, they present data descriptions that actually permit quantitative reassessments and have a history of doing so (e.g. the appendix in [8] ). This leads me to believe that they analysed their data all in good conscience and to hope that they, too, will conclude that their own data do not, in fact, support their hypothesis.

I end this paper on an upbeat note. Even though I have argued that the analytical tools employed in cph research generally leave much to be desired, the original data are, so I hope, still available. This provides researchers, cph supporters and sceptics alike, with an exciting opportunity to reanalyse their data sets using the tools outlined in the present paper and publish their findings at minimal cost of time and resources (for instance, as a comment to this paper). I would therefore encourage scholars to engage their old data sets and to communicate their analyses openly, e.g. by voluntarily publishing their data and computer code alongside their articles or comments. Ideally, cph supporters and sceptics would join forces to agree on a protocol for a high-powered study in order to provide a truly convincing answer to a core issue in sla .

Supporting Information

Dataset s1..

aoa and gjt data extracted from DeKeyser et al.'s North America study.

https://doi.org/10.1371/journal.pone.0069172.s001

Dataset S2.

aoa and gjt data extracted from DeKeyser et al.'s Israel study.

https://doi.org/10.1371/journal.pone.0069172.s002

Script with annotated R code used for the reanalysis. All add-on packages used can be installed from within R.

https://doi.org/10.1371/journal.pone.0069172.s003

Acknowledgments

I would like to thank Irmtraud Kaiser (University of Fribourg) for helping me to get an overview of the literature on the critical period hypothesis in second language acquisition. Thanks are also due to Martijn Wieling (currently University of Tübingen) for pointing out an error in the R code accompanying an earlier draft of this paper.

Author Contributions

Analyzed the data: JV. Wrote the paper: JV.

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2009 Articles

The Critical Period Hypothesis: Support, Challenge, and Reconceptualization

Schouten, Andy

Given the general failure experienced by adults when attempting to learn a second or foreign language, many have hypothesized that a critical period exists for the domain of language learning. Supporters of the Critical Period Hypothesis (CPH) contend that language learning, which takes place outside of this critical period (roughly defined as ending sometime around puberty), will inevitably be marked by non-native like features. In opposition to this position, several researches have postulated that, although rare, nativelike proficiency in a second language is in fact possible for adult learners. Still others, in light of the robust debate and research both supporting and challenging the CPH, have reconceptualized their views regarding a possible critical period for language learning, claiming that in combination with age of exposure, sociological, psychological, and physiological factors must also be considered when determining the factors that facilitate and debilitate language acquisition. In this paper, a review of literature describing the support, challenges, and reconceptualizations of the CPH is provided.

• Second language acquisition--Research
• Applied linguistics--Research
• Critical periods (Biology)
• English language--Study and teaching--Foreign speakers

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Encyclopedia of Evolutionary Psychological Science pp 1–3 Cite as

Critical Period

• Yan Wang 3 &
• Jing Guo 3  
• Living reference work entry
• First Online: 26 January 2018

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Critical period ; Crucial time ; Sensitive period

A maturational stage during the lifespan of an organism in which the organism’s nervous system is especially sensitive to certain environmental stimuli. The organism is more sensitive to environmental stimulation during a critical period than at other times during its life.

Introduction

The phenomenon of critical period was first described by William James ( 1899 ) as “the transitoriness of instincts.” The term “critical period” was proposed by the Austrian ecologist based on his observations that newly hatched poultries, such as chicks and geese, would follow the object, usually their mother, if exposed to within a certain short time after birth.

According to Lorenz, if the young animal was not exposed to the particular stimulus during the “critical period” to learn a given skill or trait, it would become extremely struggling to develop particular behavioral pattern in the later life.

A vast of existing literature has...

• Critical Period Hypothesis
• Life History Theory
• Fast Life History Strategy
• Sensitive Period
• Language Acquisition

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Department of Psychology, Fudan University, Shanghai, China

Yan Wang & Jing Guo

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Wang, Y., Guo, J. (2018). Critical Period. In: Shackelford, T., Weekes-Shackelford, V. (eds) Encyclopedia of Evolutionary Psychological Science. Springer, Cham. https://doi.org/10.1007/978-3-319-16999-6_1060-1

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DOI : https://doi.org/10.1007/978-3-319-16999-6_1060-1

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The Critical Period Hypothesis: Support, Challenge, and Reconceptualization

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Given the general failure experienced by adults when attempting to learn a second or foreign language, many have hypothesized that a critical period exists for the domain of language learning. Supporters of the Critical Period Hypothesis (CPH) contend that language learning, which takes place outside of this critical period (roughly defined as ending sometime around puberty), will inevitably be marked by non-nativelike features. In opposition to this position, several researches have postulated that, although rare, nativelike proficiency in a second language is in fact possible for adult learners. Still others, in light of the robust debate and research both supporting and challenging the CPH, have reconceptualized their views regarding a possible critical period for language learning, claiming that in combination with age of exposure, sociological, psychological, and physiological factors must also be considered when determining the factors that facilitate and debilitate language acquisition. In this paper, a review of literature describing the support, challenges, and reconceptualizations of the CPH is provided.

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