Critical period in second language acquisition: The age-attainment geometry

Abstract

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.

Full text

Critical period in second language
acquisition: The age-attainment
geometry
ZhaoHong Han* and Gang Bao
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.
KEYWORDS
ultimate attainment, critical period, second language acquisition, physics laws, energy
conservation, angular momentum conservation, inter-learner differential attainment
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:
OPEN ACCESS
EDITED BY
MatjažPerc,
University of Maribor, Slovenia
REVIEWED BY
Brian MacWhinney,
Carnegie Mellon University, United States
Dingping Li,
Faculty of Science, Peking University,
China
*CORRESPONDENCE
ZhaoHong Han,
han@tc.columbia.edu
SPECIALTY SECTION
This article was submitted
to Social Physics,
a section of the journal
Frontiers in Physics
RECEIVED 11 January 2023
ACCEPTED 02 March 2023
PUBLISHED 20 March 2023
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
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
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which does not comply with these terms.
Frontiers in Physics frontiersin.org01
TYPE Original Research
PUBLISHED 20 March 2023
DOI 10.3389/fphy.2023.1142584

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,
orwhetherthisisanaccomplishmentreservedforchildren
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 roleofnatureand/ornatureinlanguage
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]hasaptlypointedout,
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
grammaraswellasforpronunciation”([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
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].
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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<.01fortheearlyageof
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,thisclaimhasbeencontested.
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:
ϵζr
()
+Λ+η2
r2−ρ
r(1)
where ζ(r)denotes the learner’s motivational energy, r the learner’s
position in the learning process relative to the TL, ηthe distance
between L1 and TL, and ρthe 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 η2
r2, and environmental
energy - ρ
r.
The energy types included in Eq. 1are 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 η2
r2represent 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).
ϵζr
()
+∧+Ueff r
() (2)
where Ueff(r)η2
r2−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
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.
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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
r0and r0
′represent the ultimate attainments for learners with
different amounts of total energy, ϵ>0orϵ<0.
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.
ϵζ∞
()
+Λ(3)
As learning proceeds, but with rstill large (i.e., the learner still
distant from the target) and the deviation energy much weaker than
the environmental energy, η2
r2≪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.
ϵ≈ζr
()
+Λ−ρ
r(4)
As learning further progresses, the environmental energy -ρ
r
becomes dominant before yielding to the deviation energy η2
r2.
Eventually, the deviation energy overrides the environmental energy,
as expressed in Eq. 1, repeated below as Eq. 5for ease of reference.
ϵζr
()
+Λ+η2
r2−ρ
r(5)
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.
ϵΛ+η2
r2
0
−ρ
r0
(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
acentralforcefield [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
FIGURE 1
Inter-learner differential ultimate attainment as a function of different amounts of total energy: ϵ>0; ϵ<0; ϵ=ϵmin [12,13].
FIGURE 2
Geometric description of the deviation parameter η.
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Database [76], is 90.25 for Italian and English but 100.33 for
Italian and Chinese.
Figure 3 illustrates differential ultimate attainment (indicated by
r
0
) as a function of the deviation parameter η.Asηincreases, the
level of attainment is lower or the attainment is further away from
the target (r= 0).
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 ηportends 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 ηvalue 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 ηvalue 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 r0(i.e., the ultimate level of
attainment) is to the TL or the higher the ultimate attainment.
From Eq. 6the ultimate attainment of any L2 learner,
irrespective of age, can be mathematically derived:
r02η2
ρ+
4εη2+ρ2
(7)
FIGURE 3
Effective potentials U
eff
with different values of η[12,13].
FIGURE 4
Double non-linearity of r0(η)[(A):first non-linearity] and η(t)[(B):
second non-linearity] at early AoA.
3 That is when the L1 becomes entrenched.
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where εϵ–Λ(i.e., total energy minus aptitude). r0here again
denotes ultimate attainment. The upper panel in Figure 4 displays
the geometry of ultimate attainment as a function of deviation, η.
For a given child learner, ηis a constant, but different child
learners can have a different ηvalue, depending on their AoA. Herein
lies a crucial difference from adult learning where ηis 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, ηis 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)≤
ηmax. Accordingly; Eq. 7can be mathematically rewritten into (8):
r0t
()2ηt
()
2
ρ+
4εη t
()
2+ρ2
(8)
Assuming that as tgrows or as AoA increases, ηincreases 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).
ηt
()ηmax
πarctg t −a
()
+π/2

(9)
where ais 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 r0(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, ηchanging 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,dr0
dt , 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
,r0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 dr0
dt 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.
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
,itsfirst
derivative to time, dr0
dt , 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’sL1becomes
increasingly robust and developed, the deviation becomes larger,
resulting in a level of attainment increasingly away from the target
(i.e., increasingly non-nativelike).
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).
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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.
att 1
r0+1
att max
()
(10)
where att stands for level of attainment. According to Eq. 10, the
smaller the r0is, the higher the attainment is.
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 ηwith
neural plasticity or sensitivity (cf [77]). The relationship
between plasticity, p(t), and deviation function, η(t),is
expressed as (11):
pt
()1
ηt
()
+1
pmax
(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),andvice versa,as
illustrated in Figure 7.
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).Butitbeginstodecline
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 attentioninCPH/L2Aresearch.
Temporally, following the [59] conjecture, the critical period
should last through early childhood from birth to age six, and
FIGURE 6
Level of attainment as a function of AoA. FIGURE 7
Plasticity as a function of age of acquisition.
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.
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Han and Bao 10.3389/fphy.2023.1142584

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
alynchpinforageeffects,providesanexplanationforthe
differential ultimate attainment of early versus late starters.
Essentially, in early AoA, ηis 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, ηis 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, ϵ1<ϵ2<ϵ3. 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
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 η2
r2that 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.
Conclusion
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
FIGURE 8
Level of attainment for total energies 1<2<3.
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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Han and Bao 10.3389/fphy.2023.1142584

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’sview
[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
questionsinrelationtooneanother—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
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated
organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or
claim that may be made by its manufacturer, is not guaranteed or
endorsed by the publisher.
Frontiers in Physics frontiersin.org09
Han and Bao 10.3389/fphy.2023.1142584

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