1 Introduction

For cognitive functioning, usually mental models are applied for learning and problem solving of individuals in interaction with their environment; e.g., [5,6,7, 10, 12, 19, 20, 22, 24, 30, 31, 33,34,35,36,37]. As an example, mental models of devices are applied to be able to use these devices; e.g., [11, 21]. The question how mental models are developed or learnt and how to control such learning processes is an interesting and challenging one, and computational models of such processes are almost absent; e.g., [3, 8, 14]. As one of the rare exceptions, in [8] a production rule modeling format is used to simulate students’ construction of energy models in physics. In general, however, research into how mental models develop especially for learning to operate a device, is hard to find.

The current paper contributes such a computational model based on multi-order adaptive network-oriented modeling [40] and illustrated for learner-controlled mental model development while learning to drive a car. Such a driver’s mental model and how it can be developed in an effective manner can be a basis for the design of virtual pedagogical agents, and for support of the interaction between driver and the adaptive automation in cars.

Network-oriented modeling for adaptive networks [40] can be applied in general to model adaptive mental processes as interactions of mental states where their connections change based on principles of network adaptation such as Hebbian learning [16]. Learning of mental models in particular involves adaptation of these connections, but also control of this learning, which is a form of second-order adaptation. The network-oriented modeling approach from [38,39,40] covers such multi-order adaptive processes.

So, as a mental model can be modeled as a base network, learning it can be modeled as (first-order) adaptation of that base network. Moreover, the control of such learning processes can be modeled as a form of second-order adaptation, which adapts the first-order adaptation. Thus, the three-level second-order adaptive network architecture for mental model development was obtained which is presented in the current paper. It is illustrated for learning mental models by a learner-controlled combination of observational and instructional learning in the context of learning how a car works and how to drive it.

First, a brief literature overview can be found in Sect. 2. In Sect. 3, the overall design of the developed second-order adaptive network architecture is discussed, covering integration of observational and instructional learning and its control. In Sect. 4, a more detailed refinement of this general architecture is discussed to address the case study involving learner-controlled integration of observational and instructional learning. Simulations for the example scenario are discussed in Sect. 5. Section 6 is a final discussion.

2 A Brief Overview of Background Knowledge

The notion of (internal) mental model has a longstanding tradition in Cognitive and Social Sciences and in Educational Sciences; e.g., [2, 5,6,7, 10, 12, 19, 20, 22,23,24,25,26,27,28,29,30,31, 33,34,35,36,37, 46]. Within educational psychology the notion model-based learning indicates learning that occurs when people construct coherent mental models [4, 5, 9, 15, 24]. Buckley [5] formulates it as: ‘Model-based learning is a dynamic, recursive process of learning by building mental models.’ [5], p. 896. More specifically, the following elements can be considered in such learning.

Learning by observation or observational learning takes place when observation and/or imitation of others is one of the sources that help the formation of mental models. In model-centered learning, trainees watch someone else perform a target behavior and then attempt to reenact it; e.g., [4, 44]. Demonstration is an often used method in teaching new motor tasks. This particular type of learning is referred to as observational motor learning. Empirical findings show that observational motor learning improves action perception and motor execution. Mirror neurons are assumed to be responsible for the ability to learn by observing and/or imitating others as they help us understand the actions performed by others; e.g., [18, 32, 41].

Learning by instruction or instructional learning assumes that instructions from an expert instructor can be helpful. For a beginner, learning by discovery or observation may involve a great deal of trial and error; e.g., [35, 36]. Hence, along with self-learning, instructions from an expert are considered useful to build accurate and effective mental models. This notion is supported by scaffolded model-based learning in which a variety of supports like prompts, questions, hints, stories, conceptual models, visualizations are provided to assist the students’ progress during learning tasks; e.g., [17].

Learner-controlled learning for the integration of observational and instructional learning is discussed by Gibbons and Gray [13], thereby putting forward that instructions serve human learning processes best when under the control of the learner. Thus instructions do not cause learning but rather support it. The scaffolded model-based learning mentioned above supports this integration. Kozma [23] suggested that individuals do use external information sources for model construction provided in specific learning environments. Learners are sensitive to characteristics of the learning environment like availability of certain information at a given time, the structure of information and how it is introduced as well as the ease with which it can be accessed. Thus, the learner’s need for instruction and ease for acquiring it are crucial for development of accurate mental models. In guided discovery methods of learning, the learner seeks for information in the environment in order to complete the initial mental model or prior understanding. It requires the learner to be proactive and direct the learning experience. However, in expository teaching methods, an instructor directs the mental model progression by providing adequate information [29]. Meela, and Yuenyong [25] demonstrated in their study that Model-Based Inquiry (MBI) could support a student’s mental model in scientific learning. MBI focuses on developing students’ formulations of questions and procedures, creating and communicating conclusions consistent with empirical evidence [27]. Knowledge of results or feedback on performance are a significant factor in learning [1]. Many studies have established that feedback is crucial in skill acquisition [43].

Thus, in the adaptive network model introduced below, it was modeled that the learner can seek for instructions whenever it may be needed or as a feedback about what she/he has learnt by observation. This was addressed by utilizing control states for instructions on a separate level within the adaptive network model by which the learner controls the amount and timing of incoming information by seeking it only when it seems appropriate to her/him. More specifically, based on the above literature, in next section it is discussed how a learning process based on mental models can be modeled by a three-level adaptive network model. Applying a network-oriented modeling perspective for adaptive networks [40], the above literature leads to the following three different description levels that have to be addressed. First of all, the mental models themselves can be described by base networks. Next, during learning, the mental models change; this change of mental models can be described by (first-order) network adaptation. Finally, control of such a learning process is a form of adaptation of the learning process; this can be described by adaptation of the first-order adaptive network for the learning process, which is called second-order network adaptation.

3 Network Architecture for Controlled Mental Model Learning

In this section a global view on the architecture of the introduced network model for learner-controlled mental model learning is discussed. In accordance with what was concluded in Sect. 2, this architecture has to cover the following three types of processes in an integrated manner: (1) The mental models themselves described by networks, (2) Learning as change of mental models described by first-order network adaptation, and (3) Control of learning processes described by second-order network adaptation. Using the notion of multi-level network reification [38,39,40], these three description levels indeed can be modeled adequately by a three-level second-order adaptive network architecture as depicted in Fig. 1.

Fig. 1.
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Overview of the introduced second-order adaptive network architecture

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Here, for any specific application each plane contains a specific network and specific upward and downward connections define the interactions between the different levels. The more specific adaptive network model described in Sect. 4 will be such a refinement of this overall network architecture. The generic types of states and connections used at and between the three levels within this architecture are shown in Tables 1 and 2. Note that the colours used in these tables indicate to which level the states belong, as they correspond to the colours of the planes in Figs. 1, 2 and 3. At the base level, the learner’s (subjective) mental model is defined by the connections (between base states) BSX → BSY, whereas the connections (between observation states) OSX → OSY define the (objective) relations in the real world. Moreover, the connections (from observation state to base state) OSY → BSY define the mirroring process by which the observations affect the learner’s own states.

Table 1. Types of states in the introduced three level network architecture
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Table 2. Types of connections in the introduced adaptive network architecture
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Fig. 2.
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Connectivity for part of the second-order adaptive network model.

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Fig. 3.
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Connectivity for the complete adaptive network model.

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At the first reification level, the connections LWX,Y → RWX,Y and IWX,Y → RWX,Y model the integration of what is learnt by observational learning and by instructional learning, respectively. The connections ISX,Y → IWX,Y model the instruction communication actions from instructor to learner. The effect of activation of second-order state CIWX,Y is that the connection (or channel) from the instructor info state ISX,Y to state IWX,Y of the learner is opened (i.e., gets high connection weight) so that this information is transferred from instructor state ISX,Y to learner state IWX,Y.This opening of the channel ISX,Y → IWX,Y is modeled by the connections CIWX,Y → IWX,Y, where CIWX,Y represents the role of connection weight from ISX,Y to IWX,Y. Via its incoming observational learning monitoring connection LWX,Y → CIWX,Y, the control state CIWX,Y will become active depending on the corresponding LW-state LWX,Y. This models asking the instructor for verification and confirmation of what was just learnt by observation.

A more detailed display of the network’s connectivity for a specific case study can be found in Sect. 4, Figs. 2 and 3.

4 Detailed Description of the Second-Order Adaptive Network Model for a Case Study

In this section, a more detailed description is presented of the designed second-order adaptive network model applied to an illustrative case study, based on the following scenario:

Person A has almost no knowledge about a car’s components, and their interactions and how to drive a car. This person’s mental model of the car and driving it has to be learned during driving lessons. During person A’s first driving lesson, instructor B demonstrates how to start a car and get it moving. The observation of B makes that A learns an initial mental model of the car and operating it (observational learning). During A’s further learning, an iterative process of extending and/or modifying the mental model takes place, leading to a more accurate and complete mental model. Besides observational learning, also learning from instruction plays an important role (instructional learning). This instructional learning takes place by incorporating incoming information communicated by B. In this scenario this instructional learning only takes place upon request of the learner (learner-controlled instructional learning), as a form of verification and consolidation after A learnt about it by observational learning.

The network-oriented modeling approach for adaptive networks used here can be found in [38,39,40]. The characteristics used to describe networks are (for nodes or states X and Y): for connectivity connection weights ωX,Y, for aggregation combination functions cY(..) and for timing speed factors ηY. For adaptive networks the notion of network reification is used, as has been worked out in detail in [40]. This can be done iteratively to obtain multiple orders of adaptation and is applied in the way that for each state Y of the base network, for its adaptive network characteristics ωX,Y, cY(..),ηY, additional network states (called reification states or self-model states) are introduced as new nodes in the network. For example, for adaptive connectivity characteristics, states RWX,Y are added representing adaptive connection weights ωX,Y. They form a self-model of the network’s own structure in the form of a subnetwork within the network. To graphically distinguish them from states at the level of X and Y, these reification or self-model states are depicted at one level higher (e.g., see the blue planes in Figs. 1, 2 and 3 with representations of weights of adaptive connections from the base planes).

As in this case the learning is controlled, it is adaptive itself, which is depicted by the third level (purple plane) for second-order adaptation in Figs. 1, 2 and 3, which include second-order reification states CIWX,Y that represent the weight of the connection ISX,Y → IWX,Y of the middle level (see Sect. 3). In accordance with the distinction between different levels discussed above, the designed adaptive network model indeed has three levels as already depicted in Fig. 1: base level (mental model), first reification level (learning of a mental model) and second reification level (controlling learning of a mental model). Each level is graphically depicted in 3D by one horizontal plane (see Figs. 2 and 3). In these figures, the lower (pink) plane contains the base network for the mental model, whereas the middle (blue) plane represents the reification states: the IS-States, IW-states, LW-states, and RW-states, all referring to connections between the BS-states at the base level. This first reification level enables adaptation of the connections of the mental model; as discussed in Sect. 2, this is needed for learning a mental model. The structure by the lowest two (interacting) levels distinguish the two types of processes (and their interaction): using the mental model by changing the BS-states represented at the base level (used for internal simulation of the mental model) versus adjusting the mental model by changing the representations at the reification level of its connections (adaptation, learning of the mental model). The different types of states are explained in Tables 1, 3, 4. Figure 2 depicts the connectivity for only a part for a small number of the states to support better understanding. The connectivity for the complete network model is shown in Fig. 3. The second reification level (purple plane) enables to control the learning process by changing some of their intra-level connections within the first reification level (which affects the dynamics of these first-order states), based on the second-level reification CIW-states (control states); this is used to model learner-controlled instruction.

Table 3. Explanation of the base level states in the network model for the case study
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Table 4. Explanation of reification level states in the network model for the case study
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The conceptual representation of a temporal–causal network model like the one mentioned above can easily be transformed in an automated manner into a numerical representation using a dedicated modeling environment; this results in difference or differential equations ([40], Chapter 9):

$$ \begin{aligned} Y(t + \Delta t) = & Y(t) + {\varvec{\upeta}}_{Y} \left[ {{\mathbf{aggimpact}}_{Y} (t) - Y(t)} \right] \, \Delta t \\ {\text{or d}}Y\left( t \right)/{\text{d}}t = & {\varvec{\upeta}}_{Y} \left[ {{\mathbf{aggimpact}}_{Y} (t) - Y(t)} \right] \\ \end{aligned} $$
(1)

where \(\mathbf{aggimpact}_{Y} (t) = \mathbf{c}_{Y}\mathbf{(}{\varvec{\upomega}_{X_{1},Y}} {X_{1}}({t}),\ldots,{\varvec{\upomega}_{X_{k},Y}} {X_{k}}({t})\mathbf{)}\)

In the model presented here, for the states, the following combination functions were used, all generating values in [0, 1] (assuming that their argements are in [0, 1]). The Euclidean combination function eucln,λ(V1, …, Vk) where n is the order (which can be any positive real number), and λ the scaling factor is defined by:

$$ {\mathbf{eucl}}_{{n,{\varvec{\uplambda}}}} (V_{ 1} , \, \ldots , \, V_{k} ) = \varvec{\lambda} $$
(2)

where V1, …, Vk ∈[ 0, 1] indicate the impacts \(\varvec{\upomega}_{X_{i},Y} {X_{i}}(t)\) from the states X1, …, Xk from which Y has an incoming connection. The Advanced logistic sum combination function alogisticσ,τ(…) is used with steepness σ and threshold τ is defined by (with similar V1, …, Vk as above):

$$ {\mathbf{alogistic}}_{{{\varvec{\upsigma}},{\varvec{\uptau}}}} (V_{ 1} , \, \ldots ,V_{k} ) \, = \left[ {\frac{1}{{1 + {\text{e}}^{- {\varvec{\upsigma}}\left( {V_{1} + \ldots + V_{k} - {\varvec{\uptau}}} \right)}}} - \frac{1}{{1 + {{\text{e}}^{\varvec{\upsigma \uptau }})}}}} \right]\left( { 1+ {\text{e}}^{{ - {\varvec{\upsigma \uptau }}}} } \right) $$
(3)

The Hebbian learning combination function hebbμ(..) for learning of the connection from state X to state Y used in particular for the LW-states is defined by

$$ {\mathbf{hebb}}_{\varvec{\upmu}} (V_{ 1} , \, V_{ 2} ,W) \, = V_{ 1} V_{ 2} \left( { 1- W} \right) + {\varvec{\upmu}}W $$
(4)

where μ is the persistence parameter, V1 stands for state value X(t), V2 for Y(t), and W for the learnt connection weight reification state value LWX,Y(t). A more detailed specification in terms of role matrices can be found in [46].

5 Example Simulation Scenario

The computational network model was simulated using the dedicated software environment implemented in Matlab as described in [40], Ch 9, to study the development of a mental model for a car’s functioning and driving it; see Figs. 4, 5, 6 and 7. For the simulation Δt = 0.5 was chosen, the total time 800 (so 1600 simulation steps); the time scale is left abstract here, for example, it could be seconds. The speed factor for the BS-states were set at 0.4, for OS-states at 0.05, for IW-states at 0.1, and for LW-states and RW-states at 0.4. For the second-order CIW-states the speed factor was set at 0.4. The connection weights between the states and the other characteristics of the network model and the initial values are shown in [46]. For example, all BS- and OS-states use either the euclidean function or the logistic sum function, all LW-states the Hebbian learning function, the IW-states the logistic sum function, the RW-states the first-order euclidean function, and the CIW-states the logistic sum function. All BS-states have initial value 0. All OS-States have initial value 0, except the first OS-State X15 which has an initial value of 1. For all the IW-, LW-, and RW- states, the initial value was set at 0.1.

Fig. 4.
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Dynamics of the Base States X1-X14 showing internal simulation of the mental model.

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Fig. 5.
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Base States X6 (Clutch-on) and X7 (Gearbox-neutral) with impact from OS-states X20, X21and LW-state X45, RW-state X46 and learner IW-state X44 and instructor IS-state X85 with control by CIW-state X102

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Fig. 6.
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All control CIW-states showing impact from the corresponding observational learning LW-states

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Fig. 7.
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All control CIW-states showing their impact on the corresponding instructional learning IW-states

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The IS-states have constant value 1, as they refer to the knowledge of the instructor (see also Fig. 5). Note that it has been specified in such a way that only one of the IW-state or LW-state is not enough to get an RW-state with a high value close to 1. A typical pattern is that first, based on a learnt LW-state only, the RW-state gets a value somewhere in the middle of the 0–1 interval, and only after instructional learning making the IW-state high, the RW-state value increases to a high value close to 1.

The typical pattern is that first based on the value of LW-state (i.e., by observational learning), the second-order CIW-state is activated (Fig. 6) which in turn makes the IW-state getting a value close to 1 (Fig. 7). The RW-state gets a value somewhere in the middle of the 0–1 interval, and only after the learner seeks instructional information (or feedback) making the IW-state high, the RW-state value increases to 1. This shows that the learner actively engages seeking more information to confirm the accuracy of what he/she has learnt by observation. The learner hence controls the amount of information (s)he needs in addition to complete her/his learning based on her/his current level of understanding by own observation. See also Sect. 3. The results indicated in Fig. 5 display the connection between two BS-states X6 and X7. Here it can be seen that, together with the value of X6 becoming 1 at time 210, the OS-state X21 affects the value of X7 together with RW-state X46 which combines the weights of the related LW-and IW-states. The CIW-state controls the weight of IW-state according to the LW-state’s weight.

The state X7 reaches value 1 at time 250 by an S-curve. The IS-state representing knowledge of the instructor remains at 1 all the time (a knowledgeable instructor).

As a form of evaluation, in Figs. 6 and 7 it is displayed how the activation of each CIW-state follows the activation of the corresponding LW-state, and how in turn the activation of the CIW-state is followed by the corresponding IW-state. This confirms that the model displays the intended behavior that first observational learning takes place, after which there is a learner initiative to request corresponding instruction, and appropriate instructional learning indeed takes place after that.

6 Discussion

In the present paper, controlled learning of mental models was explored. The learning usually integrates different types, such as observational learning and instructional learning. For an effective learning process, appropriate timing of the different types of learning is crucial. To control this timing, the mental network adaptation process itself has to be modeled in an adaptive manner as well. The proposed second-order adaptive mental network model addresses this, where the first-order adaptation process models the learning process of a mental network model and the second-order adaptation process controls the focus and timing of the types of learning.

It was illustrated for a case study of learner-controlled mental model learning in the context of a car and driving it, where the learner is in control of the integration of observational learning and instructional learning. Based on the implemented second-order adaptive network model, the learner-controlled learning of mental models while learning about a car and how to drive it based on literature was simulated and shown to work as expected from the literature.

Much literature exists which describes the learning of mental models and is mentioned in the paper. But, formalized computational models for it are very rare; exceptions are [3, 8, 34, 42]. In [8] a production rule modeling format is used to simulate students’ construction of energy models in physics and in [34] the PDP modeling format was used to model mental models. In [42] a relatively simple adaptive God model was described, and in [3] the focus is on learning to drive a car. However, in all four cases [3, 8, 34, 42] no second-order adaptive network model is obtained, so there is no control of the learning processes. That is a main difference with the current paper, where the focus is on the control and this is fully addressed by designing a second-order adaptive mental network model. Note, however, the following disclaimer: the literature on mental models is extremely diverse, so there cannot be a claim that the way in which mental models are formalised here from a causality perspective, would be the best formalisation for all of the wide variety of suggested forms of mental models.