Introduction

Numerous theories have been proposed to describe processes that underlie human cognitive function and intelligence (e.g., Ackerman, 1996; Anderson, Bothell, Lebiere, & Matessa, 1998; Das, Kirby, & Jarman, 1975). Many models focus specifically on the construct of working memory (Baddeley, 2000; Cowan, 2005; Ericsson & Kintsch, 1995), which refers to a system of processes in the working mind that temporarily store and manipulate information in the service of cognitive goals. The majority of models agree that working memory is of limited capacity, although few can account for the mechanism that brings about this limit. Cognitive limits are better explained by theories that follow a developmental perspective, which can account for the progression and advancement of cognitive performance as a function of age (e. g., Case, 1992; Demetriou & Spanoudis, 2018; Halford, Wilson, & Phillips, 1998; Halford, Cowan, & Andrews, 2007; Pascual-Leone, 1970). This paper focuses on a dialectical-constructivist (Pascual-Leone, 1987, 2014; Pascual-Leone & Johnson, 2005) general theory of cognitive development and its advantages for predicting and interpreting cognitive performance.

The Theory of Constructive Operators (TCO) was first conceived in the 1960s and the first paper stemming from it was a mathematical model for predicting Piagets developmental stages (Pascual-Leone, 1970). The TCO is a general theory of development (Arsalidou, Pascual- Leone, & Johnson, 2010; Pascual-Leone, 1970, 1995, 1996, 2014; Pascual-Leone & Johnson, 2011; Pascual-Leone, Pascual-Leone, & Arsalidou, 2015). It was influenced by both Piagets developmental constructivist theory and the theories of Vygotsky and Luria (Pascual-Leone, 1987,1995, 1996,2012,2014; see also Miller, 2011), as well as neuropsychology/neuroscience. Central to our theory is the developmental construct of mental-attentional capacity. Mentalattention is related to Lurias (1973) functional system units 1 and 3, i.e., arousal and regulation of cortical activation tone. Explicating and combining Lurias and Piagets ideas, we state that mental/endogenous attention is a limitedcapacity brain resource that can serve to explain maturations changes in working memory (Pascual-Leone, 2000, 2019; Arsalidou et al., 2010). Based on current cognitive developmental (neoPiagetian) and neuroscientific evidence, we see unspecific arousal and automatic attention (Lurias, 1973, functional unit 1) as intertwined with, but differentiated from, intellectual/voluntary/mental (or endogenous) attention, which is a hidden specific regulation/resource operator expressed in the brain by prefrontal (excitatory or inhibitory) activity; and which is carried by prefrontal- basal-ganglia connections to the thalamus and is modulated by specific networks of slow neurotransmitters (e.g., dopamine). This mental endogenous attention is, we believe, related to and helps to explicate Lurias unit 3.

The capacity of mental attention is indexed by the number of items, not facilitated by the situation, that one can maintain and manipulate in mind (i. e., within the focal, most activated part of a mental centration). Organismically, these items are represented by schemes — informationbearing brain circuits. In addition to the schemes (subjective operators), the theory is framed in terms of hidden (not manifested in consciousness) organismic operators such

as mental attention (i. e., content-free brain specific regulations or utilities) and organizing principles (general regulations, Pascual-Leone, 1969, 1970, 1995; Pascual- Leone & Johnson, 1991,2005,2011; see Figure 1). Operators are general-purpose and content-free specific regulations that can apply on schemes in any domain, monitored by the mediation of special executive-control schemes. Psychologically, schemes are self-propelling information-bearing units that can be classified into three groups: executives, operatives, and figuratives (see Figure 1).

Our principle of Schematic Overdetermination of Performance (SOP — Pascual-Leone & Johnson, 1991,1999, 2005, 2011) expresses the spreading of activation in the brain, and can be seen as a generalization to the whole brain of th e final common path of neuronal resolution convergence originally proposed by Sherrington for the motor networks (Sherrington, 1906; McFarland & Sibly, 1975). This SOP process serves to determine which schemes eventually apply at any moment to generate an outcome. This principle stipulates that any actual performance is the product of all compatible activated schemes occurring at that time, with as many schemes possibly converging to co-determine, beyond the pragmatic need (this is overdetermination), the performance in question. Sherrington seems to have shared already this generalized principle of convergence, because he writes: “Where it is a question of mind’ the nervous system does not integrate itself by centralization upon one pontificial cell. Rather it elaborates a million-fold democracy whose each unit is a cell” (Sherrington, 1940, p. 277). Specifically, all activated and compatible schemes (because they are self-propelling — Piagets assimilation function) will apply together to co-determine performance. Mutually incompatible schemes compete, and those more strongly activated eventually apply, while other dialectically-opposing schemes are inhibited and may not apply afterwards.

Within misleading situations, where many unwanted schemes are activated, this model of endogenous mental attention could be seen as intellectual/voluntary attention:
the working mind’s spotlights, whose activation is brought about by task-relevant interacting general-purpose brain systems (hidden operators), local informational processes (schemes — subjective operators), and general regulations (organismic principles). Their dynamic synthesis generates thoughts and actions within a limited mental-attentional capacity symbolic organization. In facilitating situations, performance is easily determined by all activated schemes. However, in misleading situations mental- attentional capacity is needed to boost into dialectical dominance the task-relevant schemes, and so to synthesize the wanted performance. Mental-attentional capacity improves with age, and can be quantified within suitably misleading situations (Pascual-Leone, 1970; Pascual-Leone & Johnson, 2011).

Metasubjective task analysis can be used as a tool to interpret and predict the processing demands of a task situation (Arsalidou et al., 2010; Pascual-Leone & Johnson, 1991, 2005, 2011). We call our analytical method Metasubjective Task Analysis because it is done by adopting a perspective “from within” the person’s own processes (Pascual-Leone, 2013; Pascual-Leone & Johnson, 2017). This technique helps to estimate the overall mental demand imposed by a task; it uses functional assumptions about the repertoire of schemes and the brain resources (operators and principles) of the person who attempts to solve a task. Operators, schemes, and principles, as well as the model of endogenous mental attention and task analysis, are discussed in more detail below.

Operators, Schemes, and Overdetermination of Performance in the TCO

Operators

Operators express content-free general resources of the brain’s “hardware,” i.e., anatomical and functional-structural constraints providing specific regulations-, given suitable executive competence, they can be used by the self as resource regulations in dynamic syntheses of intended performances (Pascual-Leone, 1970; Pascual-Leone & Johnson, 1991,2005, 2011). Thus, TCO proposes operators (see Table 1) corresponding to specific regulations that explain the synthesis and emergence of novel performance. These operators are fully interpretable within the brain’s functional dynamics or regulatory processes (Pascual-Leone & Johnson, 2005; Arsalidou & Pascual-Leone, 2016). Elimination of any one of the operators should elicit an observable deficit. Because organismic operators are not content-specific, their application by the individual’s self (and their interpretation by theoreticians) is not constrained to domain-specific situations; this is a great advantage over other theories (e.g., Baddeley’s [2000] Working Memory theory).

In the notation of the TCO, operators are symbolized by abbreviations of the general functions they represent (e.g., Pascual-Leone & Johnson, 1991, 2005, 2011). Table 1 presents in a theoretically-plausible evolutionary order the operators with their functional gist, and the brain regions likely to embody them (Arsalidou, 2003; Pascual-Leone, 1989; Pascual-Leone & Johnson, 1991,2005). Operators and their evolutionary order (ontogenetic and phylogenetic) are assumed to emerge in evolution, as do all biological characteristics, following an order of survival urgency coupled with complexity: the more urgent and less complex operators should appear first. In this sense, evolutionary order can be expected to unfold from birth or early development until later years in childhood and adolescence and then regress during aging. It begins with basic affective and biological needs, and proceeds to learning by acting in the environment and observing/internalizing recurrent invariances, to finally achieve organized (goal-directed) abstract problemsolving capabilities. The empirical rationale for this order draws on phylogenetic and ontogenetic evidence (Gogtay et al., 2004; Morgane, Galler, & Mokler, 2005; Rakic, 2009). For instance, we know that the limbic system, arising from midbrain structures and serving emotional and motivated activities, is a phylogenetically older brain system when compared to the neocortex arising from the forebrain (Morgane et al., 2005). Thus, an operator regulating affects should arise first. Similarly, operators that regulate a lower cognitive level, such as space (parietal) and time (temporal) perceptual activities, should emerge developmentally before the much higher mental (prefrontal) processes, as Leibniz and Kant in some sense claimed. Effortless flow-structuring regulation (Time, T-operator) and spatial structuring (S-operator) are preconditions for knowing (as Kant knew well), and they are directly mediated by the ventral (occipito temporal) and dorsal (occipito parietal) processing streams (for reviews, see Kastner & Ungerleider, 2000; Ungerleider & Haxby, 1994).

Ontogenetic brain growth has been studied by examining the maturation of gray matter from childhood to adulthood. It demonstrates that after the maturation of sensorimotor areas, parietal and temporal areas develop prior to the prefrontal cortex, whose development continues well into young adulthood (Gogtay et al., 2004). Our evolutionary interpretation of schemes is consistent with the idea of distinct functional-hierarchical systems within the cortex (i.e., levels of areas used by Luria, 1973,1980 — primary, secondary, tertiary — relate to ways of processing complexity). Vygotsky’s and Luria’s ideas influenced the TCO (Pascual-Leone, 1987, 1996, 2014; Pascual-Leone, Johnson, Baskind, Dworsky, & Severtston, 2000). For instance, the TCO proposes different categories of schemes. Operative (executive schemes are a type of operative scheme) and figurative schemes are related respectively to frontal and posterior cortices. Such understanding of schemes was influenced by the brain organization proposed by Luria (1970). Critically, one way the TCO is unique in interpreting cognitive development is that it provides mechanisms that can explain the principles of praxis, modular organization, and dynamic synthesis of schemes, which involve both innate mechanisms and social-cultural learning (e.g., Pascual-Leone, 1996).

In a mature and efficient (i.e., adult) brain, operators and schemes engage with each other in an adaptable and flexibly hierarchical (often called heterarchical) manner. Simple situations (content) that do not require effortful mental activities may occur, and they are often coordinated by the sole means of T- and S-operators. The T-operator refers to the flowing and effortless temporal structuring of schemes’ sequences; that is, temporal representations of
internalized episodic memory of invariant sequences. These “fluent” T-structures constitute sequential chunks that often can be spontaneously learned and repeated without intervention of the mental Hf-operator defined below. For instance, when reading a text message, to understand it we activate representations of its scheme sequences (word and phrase nested meanings) effortlessly, thanks to the T-operator. Similarly, the spatial S-operator constructively abstracts from perceptual-spatial schemes, thus effortlessly creating their relations of narrative coexistence (here-and-now interrelated patterns of co-activation). For example, we easily represent the layout of a home we visit and can recognize its design in a set of architectural outlines; likewise, we can contrast and compare a large airport with another smaller one, using spatial characteristics (configural designs, dimensions, etc.), without much mental-attentional effort.

More complex situations engage additional operators, some of them (the M and I operators) mentally effortful. Cognitive actions in diverse complex situations result from coordinated schemes boosted/inhibited or specifically regulated by operators that have appropriately different regulating characteristics. For instance, if we have some mathematical training, solving a mathematical integration problem would be a complex task for most of us. We need to plan the steps to follow for reaching a solution, and we must identify the parameters involved as we ignore distractors. To regulate/coordinate such a task, a changing set of dominant executive schemes (“the executive”), monitored by an E-operator, will be needed. This E-operator produces the effective executive-organizing power (within each situation) of the currently dominant and compatible activated executive schemes within the person’s repertoire. Executive schemes can be thought of as mental strategies activated when needed for the task. These executive controls (control executives) serve to regulate the functioning of other organismic operators (such as M and I), propitiating their application on action schemes to solve the task. Complementing the control executives, task executives are executive schemes that determine which actual action schemes to use in a given task, and the order of application.

Our next operator attempts to explain mental attention — mental in the sense that it is not just perceptual or automatic attention, but effortfill internal attention (the mind’s work effort); it is an endogenous attention effortfully boosting activation of schemes relevant for the task. It boosts schemes (internal or external information) not facilitated by the situation that should be kept in mind or used in an action. According to Pascual-Leone (2019; Pascual-Leone & Johnson, 2005, 2011, in press), mental attention is best understood as a functional system constituted by four distinct hidden operators: E (executive — currently task-relevant executive schemes), M (mental-activation booster), I (mental-attentional inhibition), and F (the neoGestaltist field factor for simple syntheses — a minimization booster). These four operators in their interaction cause the emergence of mental attention. This form of attention has also been called endogenous, intellectual, voluntary, or executive attention. We prefer the term mental to executive attention, because executive processes first appear when the child is 12 months of age but mental/endogenous attention can in fact be recognized in the baby at 3 or 4 months (Hendry, Jones, & Charman, 2016; Johnson, Posner, & Rothbart, 1991).

The mental-attentional M-operator, whose capacity would represent the individual’s mental attentional capacity, can effortfully boost activation of task relevant schemes that are not sufficiently activated by other means (e. g., S- or T- or F- or А-operators). This operator provides mental- attentional energy that can be used intentionally to boost schemes in problem solving. The M-operator is a limitedcapacity resource that grows with age up to adolescence. Limitation in mental attention is often the reason complex multi-tasking is challenging, explaining in part why step-by-step approaches tend to facilitate problem solving. The attentional inhibition or I-operator (i. e., an interrupt function) causes effortful inhibition of irrelevant schemes. This inhibition operator is such an important constituent of mental attention that a leading team investigating both working memory and fluid intelligence has proposed that attentional inhibition processes (helping to disengage from outdated information) are the essential differential function of fluid intelligence (“higher fluid intelligence is indicative of self-initiated disengagement”; Shipstead, Harrison, & Engle, 2016, p. 779). They predicate this claim on the grounds that fluid intelligence is best measured using misleading problem-solving tasks that require truly novel solutions. In contrast, they argue, working memory, although much involved in disengagement, has “maintenance” as an essential function — that is, the activation boosting for relevant information. We understand these theoretical ideas to imply that mental attention is the preeminent (encompassing) causal construct for both fluid intelligence and working memory; indeed, mental attention differentiates and subsumes the activation booster of attention (M) and attentional inhibition/interruption (I). From this perspective, mental/executive attention could be understood as a superordinate causal determinant of both working memory maintenance and fluid intelligence/disengagement. Shipstead and colleagues (2016, p. 784) state: “the highest stratum is general ability (g) which we conceive of as similar to executive attention.” We basically agree with such a proposition, as well as with their claim that storage is not an essential function of working memory. Indeed, information storage of working memory is not a causal factor but a way to measure the activation-boosting capacity of mental attention (Pascual- Leone & Baillargeon, 1994; Pascual-Leone & Johnson, 2005,2011).

The F-operator stands for the automatic “simplicity” organizational principle (a minimization booster) that spontaneously occurs in the dynamics of our internal field of activation (Berthoz, 2012; Rock, 1983). It corresponds to the processes of lateral inhibition as they occur in the cortex (Edelman, 1987). F corresponds to the field factor or field effects mentioned in psychology by neo-Gestaltists and by Piaget (often also called perceptual or representational closure, Minimum principle, Stimulus-Response compatibility, etc.). This field factor plays a role in the organization of content within mental attention by bringing closure (i. e., automatic simple structuring as an organized totality) to mental representations or actions, minimizing complexity while maximizing their adaptation to given external and internal constraints. As mentioned, our

theory (TCO) explicates mental attention as the functional system of brain regulations coordinating the work of these four operators. Thus, mental attention = <E, M, I, F> (Pascual-Leone & Johnson, 2005, 2011; see Figure 2). If any of these four operators is missing, the causal-overde- termination function of mental/executive/endogenous attention will not he explainable analytically, at least within the truly-novel problem-solving tasks that express fluid intelligence.

Another set of operators can apply on activated executive and action schemes to cause learning (Pascual- Leone & Goodman, 1979; Pascual-Leone & Johnson, in press). The content-learning booster, or C-operator, corresponds to simple associative learning via classical and/or operant conditioning, and is expressed by schemes derived from this sort of simple associative content learning. The logical or L-operator is a logical-structural learning booster that often coordinates content-experiential processes with more internal (memory-based) mental processes, abstracting the relations. We call this form of learning logical because it internalizes task-essential functional invariants that constitute the actual (empirically grounded) functional-relational infrastructure of the activity from which it has been learned or (in Piaget’s sense) reflectively abstracted.

Specifically, the LC-operator refers to the automatized logical-structural learning that boosts the coordination of content learning via over-practice. The LM-operator refers to effortful logical-structural learning that results from the application of mental attention.

A critical operator in cognition is the (broadly defined) affective А-operator, which expresses the affective spontaneous boosting/inhibition of cognitive schemes that arousal of affects/emotions can provided in situations. In this paper, we subsume under A both basic affects and personal-being, affect-and-cognitive, psychosocial regulations that elsewhere Pascual-Leone has called В-operator (Pascual-Leone & Goodman, 1979; Pascual-Leone, Goodman, Ammon, & Subelman, 1978). These regulations include spontaneous self- or situation- or task-related affective-cognitive boosting (or inhibition) of schemes, as happens in spontaneous motivation and affectively driven attentive arousal elicited by situations. In psychological experiments, it is often a challenge to motivate participants, and motivation may affect the cognitive effort they use in a situation. For instance, a prize or an intrinsic motive (an affective goal) might affect motivation to engage in solving a complex integration problem. Similarly, how much we have eaten during a large dinner may change the boosting power (A-operator) of our hunger, reducing the size of cake we choose for dessert afterwards.

Schemes

Schemes are information bearers. They are dynamic and self-propelling psychological units (Pascual-Leone & Johnson, 1991, 2005, 2011), expressed in brain cell assemblies and networks (e.g., Arsalidou, Sharaev, Kotova, & Martynova, 2017; Sharaev, Zavyalova, Ushakov, Kartashov, & Velichkovsky, 2016). Schemes can be modified with experience and learning, and schemes that are applied more frequently may have a higher activation propensity (Piagets assimilation) and a higher likelihood of being engaged in their specific (cue driven) sort of relevant situations. For instance, if you usually drive in Moscow, where the cars are configured for the right-hand side, and one day you find yourself driving in London, you may reach for the handbreak on the wrong side or drive on the wrong side of the road, because in London cars are configured to drive on the left-hand side.

Cognitive schemes (i.e., information-bearing units evaluated as true, false, or uncertain) appear under three distinct categories: figurative, operative, and executive (Pascual-Leone & Baillargeon, 1994; see Figure 1): (a) figurative schemes cause perceptions and representations of concepts or objects (e.g., an apple); (b) operative schemes express blueprints for actions and specific procedures applied on objects or concepts (e.g., eating the apple); and (c) executive schemes, a subdivision of operative schemes, embody general procedures (e. g., the plan to peel, cut, and eat an apple) and apply across content domains to regulate the specific function of other operators (e. g., M- or I- or L- or А-operator) as they apply to schemes to modify dynamism or change their degree of activation (e.g., Pascual-Leone & Johnson, 1991, 2005, 2011). Locally-adaptive functional hierarchies (called heterarchies) of schemes allow for dynamic interfaces that coordinate schemes of various complexities and are context-sensitive for meaning in distinct domains. For instance, for a child an apple may refer to a good snack or, if he or she is not hungry but playful, it may serve as a ball; for a university student the word apple may refer to a fruit but more often to the brand of his or her computer. Similarly, a hammer may serve to drive in a nail, but it can also serve as a plumb if tied to the end of a cord.

Principle of Schematic Overdetermination of Performance (SOP)

In solving a task, advanced mammals set an executive goal, which mobilizes various operators to activate and adapt relevant schemes and inhibit irrelevant ones. Effortful operators (such as M or I) are limited in the number of distinct schemes they can apply to, simultaneously, within misleading situations (Pascual-Leone, 2006). In a given situation, relevant schemes compete for activation. Some schemes prevail and eventually apply to produce a behavior or mental outcome; this behavior is synthesized by the SOP principle (Pascual-Leone & Johnson, 1991). This principle biases the competitive process of dynamic dominance among schemes (which include functional, behavioral or mental, structures; i.e., complex schemes often called schemas). Activated schemes со-determine, via SOP, the performance outcome. Note that executive schemes cannot change SOP but they can, by producing activation/inhibition of action schemes, change the SOP outcome. SOP works alongside the F-operator to integrate and provide closure, so that the most information-bearing and dominant (activated) schemes can apply in the simplest way. For instance, when completing the sentence: “I like to cut bread with a ...," a perhaps large set of schemes would be cued and activated; for most of us, the strongest scheme would likely be “knife”, due to learned familiarity. Moreover, the strength of the scheme that is going to apply, often indexed in terms of reaction time, can be influenced by prior events. Priming studies show that prior events can positively or negatively influence subsequent choices (see Frings, Schneider, & Fox, 2015; Hutchinson, 2003, for reviews).

Model of Endogenous Mental Attention

Endogenous mental attention (see Figure 2), the functional system of four operators (E, M, I, F), is a complex dynamic process by which organismic operators and schemes are brought together as a working system to produce cognitive outcomes or actions. This model could also be seen by working memory theorists as an organized nesting of psychological constructs; a functional nesting in which mental attention lies within working memory, and working memory lies within the field of activated schemes, which in turn lies within the repertoire of schemes or long-term memory (e.g., Pascual-Leone & Johnson, 1991, 2005, 2011). The E-operator acts to appoint and coordinate schemes for the task; it uses M- and I-operators to suitably regulate the schemes’ degree of activation. The SOP, together with the F-operator, acts at every embedded level, as shown in Figure 2, to determine which schemes eventually will apply.

The M-operator is a limited, general-purpose resource, whose highest quantified value, its capacity, is the maximum number of distinct schemes on which the M-operator can simultaneously apply in an act of mental attention. According to the TCO, M-capacity during the symbolic mental-processing period (i.e., in and after 3 years of chronological age in ordinary children) grows on
average by one symbolic unit every two years, reaching an average of seven units in 15 to 16 year olds, matching the capacity of adults (see Table 2; Pascual-Leone, 1970, 2012, 2019; Pascual-Leone & Johnson, 2005, 2011). This growth function has been supported by extensive research (e.g., Agostino, Johnson, & Pascual-Leone, 2010; Arsalidou et al., 2010; Arsalidou & Im-Bolter, 2016; Bereiter & Scardamalia, 1979; Im-Bolter, Johnson, & Pascual-Leone, 2006; Johnstone & El-Banna, 1986; Johnson, Im-Bolter, & Pascual-Leone, 2003; Lawson, 1983; Morra, 2001; Morra, Parella, & Camba, 2011; Morra, Camba, Calvini, & Bracco, 2013; Pascual- Leone, 1970; Pascual-Leone & Baillargeon, 1994; Pascual- Leone & Johnson, 2005, 2011; Pennings & Hessels, 1996; Powell, Arsalidou, Vogan, & Taylor, 2014). Our estimate of seven units as the maximal Af-capacity for adults is contradictory to that of some current theories, which interpret the maturational mental-attentional capacity (the maximal capacity, not just what is usually employed) as 4 or 5 units in adults (e.g., Cowan, 2001; Cowan, Ricker, Clark, Hinrichs, & Glass, 2015; Halford et al., 2007). However, their developmental prediction lacks support from reviews of behavioral data of children between the ages of 5 to 12 years (e.g., Arsalidou, 2013; Pascual-Leone & Johnson, 2011; Simmering & Perone, 2013) and contradicts analyses of Piaget’s formal-operational logic tasks (Pascual- Leone, Escobar, & Jonson, 2012).

Age-appropriate performance can be measured by using tasks scaled in their Af-demand. M-demand is the minimal number of distinct schemes that participants must simultaneously activate in order to succeed in a task. Suitable scaling of difficulty level in tasks can be achieved using theory-guided metasubjective (i.e., mental or “from within”) task analysis, which yields measures of Af-capacity (Pascual-Leone, 1970, 1987; Pascual-Leone & Baillargeon, 1994; Pascual-Leone & Johnson, 1991, 2005, 2011). In children, Af-capacity normally increases with chronological age up to adolescence. For example, a task with a symbolic-processing Af-demand of four should be successfully completed by children who are 9-10 years old or older (i.e., who have an Af-capacity of

Note:                  M-capacity is represented as e+k; e represents

the sensorimotor M-capacity developed before 3 years of age (for sensorimotor processing). This e portion of M-capacity serves to hold the general executive schemes active during symbolic tasks; к represents the number of mental-symbolic, non-executive action schemes that can be simultaneously boosted or “held in mind” during task solution.

at least four symbolic schemes; see Table 2). Tasks measuring Af-capacity are called Af-measures, and metasubjective task analysis can be used to estimate their Af-demand.

Facilitating and Misleading Situations

An important distinction in the TCO refers to facilitating and misleading situations. Identifying the degree of facilitation or misleadingness can reveal which operators are going to apply. Consider a continuum, where on one end there are simple situations with clear solutions, and on the other end are complex situations with obscured solutions that must be extracted effortfully. These are respectively called facilitating versus misleading situations (Pascual-Leone, 1970, 1980, 1989). Specifically, a situation is misleading when it increases processing demand due to salient, irrelevant features (or processes) that activate schemes inducing individuals to error, relative to the intended task performance (Pascual-Leone, 1989; Pascual-Leone & Johnson, 2005).

Misleading situations often contain interfering integral features or elicit competing executive plans that require dimensional separation (using effortful processing) to achieve the intended performance (Pascual-Leone & Baillargeon, 1994). Integral (or embedding) features are features of the situation that, due to perceptual (e.g., gestalt principles) and learning processes, appear jointly integrated into a single salient object, aspect, or pattern of the situation. This can occur when we are searching for a figure in an “embedding context” (Witkin 1949; Witkin & Goodenough, 1981). In Witkin’s Embedded Figures Task (Witkin, 1950) there is a figure to be found (e. g., a triangle) embedded into the complex patterning of the item’s total figural compound. The figure to be found is embedded because each of its parts (e.g., each of the three sides of the triangle) belongs within the total compound to three distinctly different objects or parts of the compound.

Another type of misleading situation is found in the Stroop task (Stroop, 1935), where we need to avoid the automatism of reading color words and apply our attention to the ink color, naming the colors instead of reading the color words (Arsalidou, Pascual-Leone, Morris, Johnson, & Taylor, 2013). The Stroop task is well known as a measure of inhibition (see MacLeod, 1991, for a review) which is needed because reading is automatic and literate people must inhibit reading in order to attend and respond to the incongruent ink color. Indeed, in order to focus on relevant schemes in misleading situations, we must actively inhibit perceptually salient or automatized distractors, whether the distractors are irrelevant features (e.g., surrounding drawing lines in the embedded figures test) or irrelevant actions (e.g., automatism of reading color words in the Stroop task).

Moreover, competing executive plans can also create misleading situations. This is often evident when we try to multitask and do many things at once. Dual-task paradigms (see Daneman & Carpenter, 1980; Engle, 2001) were designed to do just that. During dual tasks an individual is asked to perform two tasks simultaneously, such as reading sentences and keeping track of numbers. Interference between goals from the two tasks produces a misleading executive context (Cowan & Morey, 2007).

In contrast to misleading situations, facilitating situations contain mainly task-relevant schemes that meet the needs for solving the task (Pascual-Leone & Johnson, 1991, 2005, 2011). A facilitating situation occurs when no distractors are present or when their correction is overleamed. For instance, pro-saccade tasks may be viewed as a typical facilitating situation used in experiments. During pro-saccade tasks, participants are asked to make an eye movement (i. e., saccade) in the direction of a salient visual target. Contrarily, in anti-saccade tasks participants are asked to refrain from making an eye-movement towards a visual cue (a strong tendency driven by an automatic/prewired orienting reflex) and instead to make intentional eye-movements in the opposite direction (i.e., when a visual cue appears on the right, participants must look to the left; Hallett, 1978). The latter situation contains a strong misleading component (the automatic tendency to look at the cue), which must be intentionally suppressed/inhibited. Indeed, meta-analyses of neuroimaging data show, in addition to activity in frontal eye-fields, how parietal cortices (S-operator in terms of the TCO) are responsible for saccades (Jamadar, Fielding, & Egan, 2013). In contrast, anti-saccades additionally engage prefrontal cortices — a brain region linked to executive schemes that control attentional activation and inhibition — associated with intentional-effortful cognitive control (the E-, M-, and /-operators in our theory).

The TCO, a general theory of cognitive development can be useful to psychologists, neuroscientists, and educators. Some suggest that frameworks that explicate complex cognition can provide useful suggestions for computational models and algorithm design (Schmid et al., 2011). Let us briefly comment: TCO is a model with specific mental regulations (hidden operators) and general regulations used by everyone in their working mind, including computer task designers, programmers and users when programming a task solution or using these models. Our methods might not help in finding the best program for a given task strategy, but given one such strategy they could estimate the mental difficulty of inventing (by a designer programmer) or comprehending (by a program user) the task strategy of given programs. The TCO methods could also provide heuristic ideas about how to design (or teach) one program strategy, given a certain task. Our constructs of misleading and facilitating situations, with their theoretical explanations, might provide ways for developing algorithmic model classifiers of effective complexity in programs (from both perspectives — programming and using the programs). Furthermore, the capacity limits posed in our theory by mental attention (and particularly the Af-operator) can help to evaluate difficulty for users of the computer programs as models and in implementations.

Complex psychological phenomena, such as recursive thinking (e. g., van den Bos, Rooij, Sumter, & Westenberg, 2016) may also benefit from a TCO interpretation. Recursive thinking is a complex human activity related to language, math, problem solving, social cognition, theory of mind, etc. Any complex temporal process may need recursion for its learning or modeling. The ability to consider mental states of others and to evaluate multiple perspectives improves throughout childhood and adolescence (e. g., van den Bos et al., 2016; Im-Bolter et al., 2016). Van den Bos et al. (2016) showed that performance on recursive thinking tasks increases with age; verbal abilities only partially explain this improvement, because the flow of non-verbal mentation (time-constrained mind working) is also involved. A similar conclusion was reached by Im-Bolter et al. (2016) who reveal that “theory of mind” psycho-social reasoning depends greatly on the maturity level of the M- and I- operators formulated by the TCO perspective.

Recall that facilitation and misleadingness occur in a dynamically-graded continuum (i.e., a situation can be more facilitating or more misleading), which considerably changes the organismic operators needed to solve the task. Indeed, all things equal, successful completion of misleading tasks demands much more E-, M-, and I-capacity than facilitating tasks. How then to determine whether a task is misleading or facilitating? To identify these features and processes, we need to do a mental-process (metasubjective) task analysis.

Metasubjective Task Analysis (MTA)

This is a theory-guided, rationally-based approach used to estimate the Af-demand of a task. As mentioned, it is called metasubjective because it attempts to describe task-solution processes “from within”; that is, from the perspective of the person’s processes themselves. The theory of constructive operators provides constructs and tools to break down a task to its basic process components (schemes and organismic functions; i.e., operators and principles) that influence the task’s E-, M-, /-, and other demands (Pascual-Leone & Johnson, 1991, 2005, 2011, in press). In other words, this analysis can be used to identify an appropriate strategy for the task, and to model operative schemes (including their parameters), figurative schemes, and corresponding operators likely to intervene in the solving process. This method is particularly valuable when designing new measures for estimating/assessing individuals’ performance capabilities, such as in children and in clinical populations. For instance, to know that a symbolic-processing task requires effortful coordination of one operative and three figurative schemes leads us to expect the symbolic Af-demand of this task to be at most four units (if there is no other complicating, distracting, or misleading factors). In making the prediction, we can then anticipate that this task would be easily accessible to ordinary adults, whose functional (i. e., commonly used) Af-capacity tends to be 4 or 5, but whose structural (maximal) Af-reserve is 7 units (Pascual-Leone, 1970,2012, 2019; Pascual-Leone & Johnson, 2005, 2011). In contrast, children younger than nine years old can be expected to have difficulty in successfully solving such a task, unless facilitating factors exist, due to experience or situational characteristics.

We illustrate this MTA with a developmental task, the Color Matching Task (CMT; Arsalidou et al., 2010). We focus here on the Clown version. In this task participants see, one by one, a series of items (always a clown); for each one, they must indicate whether relevant features (i.e., colors) of the current item match those of the immediately preceding item. In other words, participants

respond to whether the relevant set of colors is the same or different in contiguous items. The colors blue and green are always irrelevant, and relevant colors vary in number from one to six, depending on the item’s difficulty level. Therefore, we have one operation (the operative scheme to scan and identify) that is applied to the relevant colors (figurative schemes) that range across items from one to six. In addition to irrelevant colors, participants need to ignore color location, as well as the face of the clown and the integral compound figure of the clown itself with its body parts (i.e., gloves, buttons, collar, etc.). These distracting/misleading factors are critical to estimate the task’s Af-demand, because misleading schemes (or features) force participants to adopt bypassing (error-avoiding) strategies that usually involve a higher number of relevant schemes — a higher Af-demand. In the present task, to Scan-Identify-Match the sets of colors becomes more complex: an additional scheme is required to effortfully extract, at every turn, one relevant target color from the clowns current item (its total set of color features, some of which are misleading or irrelevant) to be matched with the previous items set ofn (relevant) criterion colors. If we want to quantify the Af-demand for an item with n relevant colors, the estimate should be n + 2 operative schemes, where 2 stands for the Scan-Identify-Match operative plus the scheme for extracting new target-colors from the clown figure in the current item in question. For instance, a task with two relevant colors will have an Af-demand of four units. Since we know that children of 9-10 years have a symbol-processing Af-capacity of four (see Table 2), they should be able to pass this level of task difficulty, but younger children should fail. Children are assigned an Af-capacity score corresponding to the Af-demand of the highest item level they can pass reliably.

As illustrated in Figure 3, data from two empirical cross-sectional studies exhibit close correspondence between children’s obtained CMT Af-scores and the predicted theoretical Af-capacity based on age, over the whole developmental range (Arsalidou et al., 2010; Powell et al., 2014; see Figure 3). These empirical results (along with many different results across age-group samples and across tasks — some were referenced above) offer strong construct validity to this constructivist-developmental modeling which was first proposed over forty years ago (Pascual-Leone, 1970; Pascual-Leone & Baillargeon, 1994; Pascual-Leone & Johnson, 2005, 2011). The similarity of the obtained quantitative scores across types of tasks and across age-group samples speaks to the developmental reliability and construct validity of this Af-measurement method (a method that is remarkably culture cair; Arsalidou & Im-Bolter, 2016; Miller, Pascual-Leone, Campbell, & Juckes, 1989; Miller, Pascual-Leone, & Andrew, 1992; Pascual-Leone et al., 2000).

Conclusion

We have summarized very briefly the fundamental constructs, models, and quantitative predictions of the Theory of Constructive Operators and given experimental and everyday examples as simple illustrations. Originally inspired by the work of Piaget and by Goldstein’s (2000/1934) and Luria’s neuropsychology (among others), our distinct contribution has been to clarify, by way of constructivist developmental research, the intertwining of Luria’s modes of processing — his Unit 1 for regulating tone and the waking and mental states of arousal, vigilance, and attentional inhibition with his Unit 3 for programming, regulating, and verification of activity (operative and executive processes). We clarified their intertwining by explicating concepts, in particular schemes and hidden operators, with a focus on mental attention. Mental attention was formulated (but not causally explained) by William James and Luria as voluntary/intellectual attention, which we have quantified via developmental complexity analysis of many age-group samples. We have also intimated how, in some sense, mental-attention relates to consciousness and intelligence.

We highlight three other critical benefits that this theory offers.

First, a distinction is made, and the coordination predicated, between (hidden) organismic operators (specific regulations fully interpretable within the brain’s functional dynamics or regulatory processes — an issue beyond the paper’s scope) and the information-bearing schemes (information carriers, brain cell assemblies and information networks). Notice that the content-free and specific-regulation function of the hidden operators allow us to apply this modeling to explain context-sensitive functioning of schemes in many distinct content domains. This is critical, because mental-attention is a cognitive resource implicated in all sorts of problem-solving across domains and across age groups (see Arsalidou & Im-Bolter, 2016; Onwumere & Reid, 2008, for reviews).

Second, there are now well-established quantitative predictions (first introduced by Pascual-Leone in 1970) about the mental-attentional capacity characteristic of Piagetian and neoPiagetian developmental stages. This permits, via metasub)ective analysis, quantitative performance predictions across many tasks, for children of different ages. These expectations have been and can be tested developmentally.

Third, a methodology now exists for identifying a-priori the effective developmental complexity, or mental (M-) demand, of most cognitive tasks or items; this allows for rigorous age-appropriate performance predictions.

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