An adaptive network model covering metacognition to control adaptation for multiple mental models

Learning processes can be described by adaptive mental (or neural) network models. If metacognition is used to regulate learning, the adaptation of the mental network becomes itself adaptive as well: second-order adaptation. In this paper, a second-order adaptive mental network model is introduced for metacognitive regulation of learning processes. The focus is on the role of multiple internal mental models, in particular, the case of visualisation to support learning of numerical or symbolic skills. The second-order adaptive network model is illustrated by a case scenario for the role of visualisation to support learning multiplication at the primary school. (cid:1)


Introduction
Metacognition (Darling-Hammond, Austin, Cheung, & Martin, 2008;Shannon, 2008;Mahdavi, 2014;Flavell, 1979;Koriat, 2007;Pintrich, 2000) is a form of cognition about cognition. In (Koriat, 2007) it is described as what people know about their own cognitive processes and how they put that knowledge to use in regulating their cognitive processing and behavior. A sometimes used closely related term is self-regulation and when the cognitive processes addressed by metacognition concern learning, the term self-regulated learning is used. For example, in (Pintrich, 2000), self-regulated learning is described as an active, constructive process whereby learners set goals for their learning and then attempt to monitor, regulate, and control their cognition, motivation, and behavior, guided and constrained by their goals and context.
In learning, often different mental models play a role; e.g., (Gentner & Stevens, 1983;Greca & Moreira, 2000;Skemp, 1971;Seel, 2006). A specific case where the role of metacognition in learning processes is considered within educational science, is the use of multiple mental models such as in visualisation to support learning of more abstract (numerical or symbolic) skills; e.g., (Bruner, 1966, pp. 59-68;Du Plooy, 2016). An important metacognitive control decision in this context is whether or not and when to switch from one mental model to another one. In the educational science literature, much more can be found on this case, particularly for learning arithmetic or algebraic skills in primary or secondary schools supported by visualisation; see also (Bruner, 1977;Bidwell, 1972;Day & Hurrell, 2015;Freudental, 1973, Freudental, 1986Koedinger & Terao, 2002;Larbi & Mavis, 2016;Lovitt, Marriot, & Swan, 1984;Renkema, 2019;Roberts, 1989;Rivera, 2011a,b).
From a network-oriented modeling perspective, learning is usually described by adaptive mental (or neural) network models, where some of the network characteristics such as connection weights or excitability thresholds change over time. If, in addition, metacognition is used to regulate or control the learning, this implies that the adaptation (by learning) of the mental network is itself adaptive as well, which is called second-order adaptation. Thus, a network model for such processes has to address such complex structures and behaviour. In the current paper, using the modeling approach for higher-order adaptive networks from (Treuró , 2018;Treur, 2020a;Treur, 2020b), a second-order adaptive mental network model is introduced for metacognitive regulation of such learning processes. Here, the focus is on the role of multiple mental models in case of visualisation to support learning of more abstract (numerical or symbolic) skills. The adaptive network model is illustrated for a case study on the role of visualisation to support learning multiplication at the primary school as described, for example, in (Bruner, 1966;Day & Hurrell, 2015;Du Plooy, 2016;Freudental, 1973, Freudental, 1986.
In this paper, first in Section 2 more background knowledge is discussed on metacognition and the role of visualisation in learning processes. In Section 3 the networkoriented modeling approach used is briefly explained. Next, in Section 4 the introduced second-order adaptive network model is described in some detail. In Section 5, it is shown how this model was used to perform simulations for the illustrative example scenario. Finally, Section 6 is a discussion.

Metacognition and multiple mental models
Literature on metacognition, sometimes also called self-regulation, can be found, for example in (Darling-Hammond et al., 2008;Shannon, 2008;Mahdavi, 2014;Flavell, 1979;Koriat, 2007;Pintrich, 2000). The focus is here on the role of metacognition in learning. For example, in (Pintrich, 2000, pp. 452-453) the following assumptions for self-regulated learning are described: It is a process whereby learners set goals for their learning and monitor and control their cognition, motivation, and behavior guided by these goals. Learners actively construct their own meanings, goals, and strategies. Learners can monitor, control, and regulate certain aspects of their own cognition, motivation, and behavior, and some elements of their environment. Some type of criterion or standard is used to assess whether the process should continue as is or if some type of change is necessary.
Self-regulatory activities are mediators between personal and contextual characteristics and actual performance.
In line with these assumptions, in (Pintrich, 2000, pp. 453-461, p. 454), the following phases for self-regulation are described: Cognitive planning and activation Cognitive monitoring Cognitive control and regulation Cognitive reaction and reflection In (Koriat, 2007, p. 290), metacognition is described by what people know about cognition and in particular their own cognitive processes, and how they use that in regulating their cognitive processes and behavior. Assumptions mentioned there are: Self-controlled cognitive processes have measurable effects on behavior (Koriat, 2007), pp. 292-293. Feelings, such as the feeling of knowing are part of monitoring, and exert a causal role on the control of cognitive processing (Koriat, 2007), p. 293, p. 314-315. There is a causal relation from monitoring to control (Koriat, 2007), p. 315.
So, in both descriptions of Pintrich (2000) and Koriat (2007) on metacognition (as well as in most other literature on metacognition), monitoring and control of the own cognitive processes are central concepts (where Koriat also emphasizes the feeling or experiencing that comes together with monitoring). These processes work through a causal cycle where the own cognitive processes affect the metacognitive monitoring, this monitoring in turn affects the metacognitive control, and this control affects the own cognitive processes. This causal cycle will indeed be incorporated in the adaptive network model introduced in Section 4. Note that metacognitive monitoring is usually based on forming and maintaining a self-model describing a (subjective) estimation of some relevant aspects of the own cognitive processes.
In the area of learning using multiple mental models (Gentner & Stevens, 1983;Greca & Moreira, 2000;Skemp, 1971;Seel, 2006), metacognition plays an important role for the decisions about when to switch from one mental model to another one. In particular, this takes place when learning numerical or symbolic skills in arithmetic or mathematics is supported by visualisations; e.g., see (Bruner, 1966;Bruner, 1977;Bidwell, 1972;Day & Hurrell, 2015;Du Plooy, 2016;Freudental, 1973, Freudental, 1986Koedinger & Terao, 2002;Larbi & Mavis, 2016;Lovitt et al., 1984;Renkema, 2019;Roberts, 1989). Here, when at some point during working with a numerical or symbolic mental model, a learner monitors that the cognitive processes get stuck, the control decision can be made by the learner to switch to working with a mental model based on visualisation, after which the outcomes can be fed back to the numerical or symbolic mental model. Within the literature in educational science as mentioned above, it is extensively described how such a detour via a visualisation can support the learning of numerical or symbolic skills. This type of use of metacognition for using multiple mental models is the main focus in the current paper.

Higher-order adaptive network models
In this section, the network-oriented modeling approach used is briefly introduced. Following (Treur, 2016;Treur, 2020b), a temporal-causal network model is characterised by (here X and Y denote nodes of the network, also called states):

Connectivity characteristics
Connections from a state X to a state Y and their weights x X,Y Aggregation characteristics For any state Y, some combination function c Y (..) defines the aggregation that is applied to the impacts x X, Y X(t) on Y from its incoming connections from states X Timing characteristics Each state Y has a speed factor g Y defining how fast it changes for given causal impact.
The following difference (or differential) equations that are used for simulation purposes and also for analysis of temporal-causal networks incorporate these network characteristics x X,Y , c Y (..), g Y in a standard numerical format: for any state Y and where X 1 to X k are the states from which Y gets its incoming connections. Within the software environment described in (Treur, 2020b, Ch. 9), a large number of around 40 useful basic combination functions are included in a combination function library.
The above concepts enable to design network models and their dynamics in a declarative manner, based on mathematically defined functions and relations. Realistic network models are usually adaptive: often not only their states but also some of their network characteristics change over time. By using network reification, a similar networkoriented conceptualisation can also be applied to adaptive networks to obtain a declarative description using mathematically defined functions and relations for them as well; see (Treur, 2018;Treur, 2020a;Treur, 2020b). This works through the addition of new states to the network (called reification states or self-model states) which represent (adaptive) network characteristics. In the graphical 3D-format as shown in Section 4, such additional states are depicted at a next level (called reification level), where the original network is at the base level. As an example, the weight x X,Y of a connection from state X to state Y can be represented (at a next reification level) by a reification state named W X,Y (objective representation actually used) or RW X,Y (subjective representation for a self-model). Similarly, all other network characteristics from x X,Y , c Y (..), g Y can be made adaptive by including reification states for them. For example, an adaptive speed factor g Y can be represented by a reification state named H Y and an adaptive excitability threshold parameter s Y can be represented by a reification state named T Y .
As the outcome of network reification is also a temporal-causal network model itself, as has been proven in (Treur, 2020b, Ch 10), this reification construction can easily be applied iteratively to obtain multiple reification levels. In the current paper, multi-level network reification will be applied to obtain a second-order adaptive mental network model addressing metacognitive control of learning in a multiple mental models context.

A mental network model for metacognitive control of learning from multiple internal mental models
In this section, the adaptive mental network model for metacognitive control on learning using multiple mental models is introduced. This adaptive mental network model has processes at three levels: The base level network for the (multiple) internal mental models used The first reification level for the learning of the internal mental models by adaptations of them The second reification level for control by adaptation of the first-order network for the learning These three levels of processes have been modeled by a second-order adaptive network based on multi-level network reification (Treur, 2018;2020a;2020b) briefly described in Section 3; the connectivity of this network model is depicted in Fig. 2. The states used are explained in Table 1. For the example mental models at the base level, on the left hand side in Fig. 2 an internal numerical mental model for an arithmetic task is included and on the right hand side a visual, geometrical mental model for it. The example task is to show (in the numerical representation) for certain given natural numbers a, b and c that The detour via visualisation considers two rectangles with vertical dimension a and horizontal dimensions b and c and their areas that are together equal to the area of a rectangle with vertical dimension a and horizontal dimension b + c, as shown in Fig. 1.

Network Characteristics: Connectivity and timing
At the base level, for the numerical mental model, the base states N 1 , N 2 and N 3 represent the given numbers a, b, and c. Base states P 12 and P 13 represent the products a*b and a*c, respectively, whereas state S 12 represents the sum b + c. Finally, base state SP 1213 represents the sum of P 12 and P 13 which is a*b + a*c, while base state PS 123 represents the product of N 1 and S 23 which is a*(b + c).
For the geometric mental model, base states RD vert , RD hor1 , RD hor2 , and RD hor3 represent the vertical and horizontal dimensions of the rectangles in Fig. 1, respectively. Moreover, RA 1 , RA 2 and RA 3 represent the areas of the three rectangles with horizontal dimension b, c, and b + c, respectively, and RA 12 the area of the two smaller rectangles together.
At the first reification level, the learning of the adaptive connections of the numerical mental model is modeled by the W-states and as input for the self-model for the metacognitive monitoring the learnt relations as estimated by the learner are represented by the two (subjective) RWstates X 23 and X 24 . Moreover, the WRD-states X 25 to X 27 model the adaptive connections from the numerical mental model to the geometric mental model used to dynamically switch from one to the other; this is part of effectuating the metacognitive control.
At the second reification level, the self-model for the status of the learning (for the own estimated learnt numerical and geometric skills) for the metacognitive monitoring is represented by the two RS-states X 28 and X 29 and the metacognitive control decisions (to switch to the geometric mental model) are modeled by the CWRD-states X 30 to X 32 , based on the impact from the self-model obtained by the metacognitive monitoring. Table 1 The states in the adaptive network model. There are two types of connections: intralevel connections (in Fig. 2 depicted in black) and interlevel connections (depicted in blue for upward and in pink for downward). At the base level, within each of the two mental models, the connections define these mental models by their internal causal impacts. For example, the connections N 1 ? P 12 and N 2 ? P 12 define that within the numerical mental model the product of a and b represented by base state P 12 depends on base states N 1 and N 2 representing these numbers.
In addition, at the base level a number of connections define how the two mental models relate to each other. For example, the connection N 1 ? RD vert from the numerical mental model to the geometric mental model defines that the vertical dimension of the rectangles within the geometric mental model depends on the number a represented by numerical state N 1 . Moreover, a connection back from the geometric to the numerical mental model such as RA 12 ? SP 1213 defines the influence of the outcomes of the geometric process on the numerical process as a form of reinforcement to amplify the learning of the numerical mental model.
The upward connections to the first reification level Wstates provide impact to the W-states so that they can adapt over time, which is modeled according to a qualitative Hebbian learning (Hebb, 1949) principle specified by (6) below. For example, connections N 3 ? W P313 and P 13 ? W P313 provide impact to W P313 so that W P313 can adapt over time. On the other hand, the downward connection from a W-state makes that the value of it is actually used in the processing of the mental model. For example, the connection W P313 ? P 13 takes care for this for W P313 so that for the weight of the connection N 3 ? P 13 the value of W P313 is used. Furthermore, the upward connections to the first reification level RW-states make that a representation for the status of some connections of the numerical mental model is formed and maintained. This is a first step toward a self-model which is the basis of the metacognitive monitoring of the own cognitive processes.
At the second-order reification level, based on impact from the RW-states at the first reification level, the selfmodel is formed and maintained by the states RS num and RS geo . Via their outgoing connections, the states RS num and RS geo of this self-model have their impact on the control decisions modeled by the CWRD-states. By their downward connections, the CWRD-states for control decisions determine the incoming connections to the corresponding WRD-states, so that the control decision is executed by realising that these WRD-states get values 1. In turn, once the WRD-state has a value 1, it makes that at the base level the corresponding connection from numerical mental model to geometric mental model is 1, which then leads to the geometric mental model states RD vert , RD hor1 , and RD hor2 getting the appropriate values from states N 1 , N 2 , and N 3 of the numerical mental model. In Fig. 3 the complete role matrix specification of the connectivity and timing characteristics of the designed adaptive network model can be found. Here in each role matrix, each state has its row where it is listed which are the impacts on it from that role. Role matrix mb lists the other states (at the same or lower level) from which the state gets its incoming connections, whereas in role matrix mcw the connection weights are listed for these connections. Note that nonadaptive connection weights are indicated by a number (in a green shaded cell), but adaptive connection weights are indicated by a reference to the (reification) state representing the adaptive value (in a peach-red shaded cell). For example, state X 5 (=P 12 ) has incoming connections from X 1 (=N 1 ), X 2 (=N 2 ), and X 13 (=RA 1 ) with connection weights represented by X 17 (=W P112 ) and X 18 (=W P212 ) and 1, respectively. These two adaptive connection weights model the reinforced (by RA 1 ) hebbian learning. Also, the states RD vert , RD hor1 , RD hor2 for the dimensions of the rectangles in the geoometric mental model have adaptive connection weights. These adaptive connections are used to model the metacognitive control of the switch from numerical mental model to geometric mental model: if the control decision is made to switch, then these connection weights (represented by the WRDstates) quickly become 1 to transfer the numbers a, b and c to the geometric mental model. This rapid transition is specified in role matrix ms for the timing, where it is indicated that the speed factors of the WRD-states X 25 to X 27 are adaptive and immediately change from 0 to 1 as soon as the CWRD-states X 30 to X 32 for metacognitive control at the second reification level change to 1.

Network Characteristics: Aggregation
The network characteristics for aggregation are defined by the selection of combination functions from the library and values for their parameters. First the six combination functions used for the model are specified by mcf ¼ ½1 2 3 9 22 23 4 ¼ ½eucl alogistic hebbqual complement À id product max Àcomposition Here the numbers are the numbers of the listed functions in the library. Next, it is specified which state uses which combination function. This can be seen in role matrix mcfw in Fig. 4.
The combination functions from the library used in the introduced network model are defined as follows: The Euclidean combination function eucl n,k (V 1 , . . ., V k ) is defined by where n is the order and k a scaling factor and V 1 , . . ., V k are the impacts from the states from which the considered state Y gets incoming connections. Note that if both parameters have value 1, then this is just the sum function and when there is only one incoming connection the identity function. This is always the case in the current model, as can be seen in role matrix mcfp. The product combination function product(V 1 , V 2 ) is defined by The advanced logistic sum combination function alogistic r,s (V 1 , . . ., V k ) is defined by: where r is a steepness parameter and s log a threshold parameter and V 1 , . . ., V k are the impacts from the states from which the considered state Y gets incoming connections The qualitative hebbian learning combination function hebbqual l (V 1 , V 2 , W) is defined by where l is a persistence parameter, W represents the weight of their connection, and V Ã i is 1 if V i > 0.1 and else 0 (here V 1 , V 2 are the activation levels of the connected states) The complemental identity combination function complement-id(V) is defined by where V is the incoming impact from a connected state The max-composing combination function maxcomposition m,n (V 1 , V 2 , V 3 ) is defined by max À composition m;n ðV 1 ; V 2 ; V 3 Þ ¼ maxðbcfðm; ½1 1; ½V 1 ; V 2 Þ; bcfðn; ½1 1; ½V 3 Þ ð8Þ Where bv/ (i,p,x) is the i th basic combination function from the library. This function composes two other combination functions from the library by using the maxfunction. It is actually defined as a special case using a The combination function eucl n,k (. . .) for n and k both 1 is used to model addition, product(V 1 , V 2 ) to model multiplication, hebbqual l (V 1 , V 2 , W) to model learning of arithmetic operations, and alogistic r,s (V 1 , . . ., V k ) and complement-id(V) to model internal metacognitive monitoring and control states for the learning. The combination function max-composition m,n (V 1 , V 2 , V 3 ) is used to reinforce the learning in the numerical mental model through the outcomes from the geometric mental model.

Example simulation scenarios
In this section, simulations of two example scenarios will be discussed to illustrate the introduced second-order adaptive network model. Both scenarios address the example task discussed in Section 3 (see also Fig. 1) for a = 2, b = 3, c = 2, which are used as constant values for base states N 1 , N 2 , and N 3 , respectively. The first scenario shows how someone who has good arithmetic skills addresses the task, without involving any switch to the geoemetric mental model; see Fig. 3. As can be seen, as one of the first, state S 23 comes up which determines the sum of N 2 representing b and N 3 representing c, which correctly ends up in value 5 (the blue line). At about the same time state P 12 (the red line) for the product of N 1 and N 2 respresenting a and b comes up, correctly ending up at 6. Similarly, P 13 (the bluegreen line) for the product of N 1 (for a) and N 2 (for c) correctly reaches 4.
Next, PS 123 of N 1 and S 23 representing the product of a and b + c is determined, which correctly ends up in 10 (the light dark green line). The determines the left hand side of the Eq. (2). At the same time, the right hand side of (2) is addressed. Therefore, SP 1213 (again the dark green line) for the sum of P 12 and P 13 comes up and correctly reaches 10. This shows that the right hand side of (2) is indeed equal to the left hand side of (2), what solves the task. In the meantime it can be seen in Fig. 3 that the self-model about the numerical mental model is formed: the two lines for the two RW-states all end up at 1, and also based on them the third (orange) line for RS num , which as a form of metacognitive monitoring tells the learner that the arithmetic skills are OK. Therefore, in this case no control decision to switch to the geometric mental model is made, and also no further learning is needed.
The second scenario is the more interesting one (see Fig. 4). Here the learner has still good arithmetic skills (connection weights 1) to address the left hand side of (2), but not for the right hand side (connection weights are only 0.1). Therefore the light brown and purple lines in the upper graph in Fig. 4 are the same as in Fig. 3, but not the lines for P 12 , P 13 , and SP 1213 needed for the right hand side of (2). Because that side gets stuck, and the self-model used for monitoring has low values showing a lack of arithmetic skills, the control decision is made to switch to the geometric mental model: all three CWRDstates come up soon and reach 1 shortly after time 5 (the purple line in the lower graph of Fig. 4). As a consequence, to execute this control decision, the WRD-states become 1 around time 5 (the red line in the lower graph of Fig. 4).
Because of that the RD-states representing the dimensions of the rectangles get their values 2, 3, and 5. Based on these, the RA-states for the areas of the rectangles are determined and get their values 4, 6 and 10. As these RA-states provide a reinforcing impact on the states P 12 , P 13 , and SP 1213 in the numerical mental model, it can be seen that with a small delay the latter states follow the RA-states to also reach values 4, 6 and 10 (the red, light blue, and dark green line in Fig. 4, upper graph). In the lower graph of Fig. 4 it can be seen what happens further concerning the adaptation levels. The lines starting at 0.1 are the W-states, and it is shown that after time 6 they start to increase to finally reach values close to 1. This is the reinforced hebbian learning process for the numerical mental model: reinforced by the inpact from the geometric mental model. Also the two RW-states and state RS num for the self-model for the numerical mental model, starting at 0.3, 0.5 and 0.4, increase after time 6. Note that the RS geo (light green line with peak near 0.9) also increases thereby supporting the decision to switch to the geometric mental model, but later on (after time 15) goes down just like the CWRD-states for the control themselves do (after time 25), as after learning the full arithmetic mental model, by the monitoring via the self-model the learner feels that there is no reason anymore to consider switching to the geometric mental model.

Discussion
Learning processes can be described by adaptive mental (or neural) network models. If metacognition is used to regulate learning (Pintrich, 2000), the adaptation of the mental network becomes itself adaptive as well, so then it involves second-order adaptation. In this paper, a second-order adaptive mental network model was introduced for metacognitive regulation of learning processes using multiple internal mental models.
It was shown how a second-order reified network model provides adequate means to model the different aspects that make the addressed topic complex: the network has a self-model about its own structure, it models mental models and their adaptation for learning, and it models dynamic metacognitive control of this adaptation. The model was applied to simulate some example scenarios that illustrate what the model does. In further work other scenarios can be addressed as well.