Mathematical models have played a critical role in human–computer interaction research for decades. For example, Fitts’s law, which quantifies the difficulty in target selection, has played a pivotal role for the development of input devices, such as a keyboard, a mouse, a joystick, and many other graphical user interfaces (e.g., menu, taskbar) [
1,
2,
3,
4,
5]. To model human psychology, such as personality, attitude, and mindset, what mathematical model can be applied? This study focuses on “mindsets”—people’s tacit beliefs about attributes [
6]—and investigates whether (1) mindsets can be extracted from a motion trajectory in target selection and (2) a dynamic state-space model, such as a Kalman filter [
7], helps quantify mindsets.
1.1. Mental State Assessment
Much research in mental state assessment has been conducted under the banner of passive Brain Computer Interface and Affective Computing [
8,
9]. Among the most well studied is mental workload. Mental workload (or cognitive load) refers to the mental costs of carrying out a task; it is determined by external (task difficulty, priority) and internal factors, attention, memory, stress, motivation and mindsets [
10,
11,
12,
13]. Mental workload is often measured by task performance (accuracy and response time), self-report (questionnaires, e.g., NASA Task Load Index—NASA-TLX [
13]) and physiology (heart rate and heart rate variability, pupil dilation, eye movement, and brain activity electroencephalography/event-related potential (EEG/ERP)). Task performance and self-report cannot deliver continuous monitoring. Physiological measures, especially brain activity, provide the most viable continuous assessment [
11].
Among many brain activity measures, the most well studied and realistic measures are EEG and ERP. Studies have shown that spectral powers of the alpha (7–14 Hz), theta (4–7 Hz) and beta (12.5–30 Hz) bands are related to cognitive load. Sterman and Mann [
14] showed that when aircraft pilots were maneuvering a difficult task, the power of the alpha band decreased. Sterman and Mann demonstrated that task difficulty in U.S. airline pilot and spectral power of the alpha wave is inversely related. Brookings et al. also showed that air traffic controller’s control responsiveness was inversely related to the power of the alpha band. Hoogendoorm et al. [
15] further showed that high workload (high attention) is correlated with high beta (12–30 Hz), low alpha (8–12 Hz), and low theta (5–8 Hz) spectral power, although the interpretation of theta is not unequivocal. Another important measure of mental workload is ERP (event related potential). Among many ERP components, P300 and ERN (error-related negativity) provide the most reliable biomarkers for mental workload assessment [
16,
17,
18,
19]. Recently, more sophisticated algorithms, such as adaptive neural network [
20] and sparse representation-based EEG signal analysis [
21,
22] have proved effective for mental assessment involving cognitive workload, emotional states, as well as brain impairments.
Despite the recent developments in mental state assessment, nearly all the aforementioned EEG/ERP studies have been conducted in tightly controlled laboratory settings with multiple electrodes (32 or more) wired to a heavy device. These EEG systems are not always practical in real world situations in which people interact constantly. Moreover, few studies have investigated the mental state analysis beyond cognitive workload and emotional states. Some other psychological variables, such as mindsets, can be studied with other means beyond EEG.
1.2. Mindsets in Motor Control
Mindsets here refer to individuals’ conceptualization of attributes, specifically the extent to which individuals view abilities as fixed or malleable. Dweck and colleagues show that mindsets influence a wide range of goal-directed behaviors including adolescents learning advanced math, athletes training for competition, business managers honing managerial skill, or college students developing interpersonal competence [
23,
24,
25,
26,
27,
28,
29]. In a computer-assisted collaborative working environment, evidence shows that growth- or fixed-mindsets affect learners’ product-acceptance and usability ratings [
30].
Mindsets modify goal-setting, task engagement, planning, feedback seeking and outcome attribution [
6]. For example, if one believes that math talent is fixed (e.g., a person was born with a “math gene” and the talent remains fixed throughout), she strives to “show off” her competence if she believes she has it, or hide it if she thinks she does not have any. In contrast, if one believes that math talent is malleable (e.g., a “math talent” can be developed through practice), then she is more likely to nurture it. In this manner, mindsets—beliefs about abilities—influence our behavior profoundly.
People’s mindsets can be reflected even in a simple motor task, such as selecting and pressing a button on the computer screen. In cognitive science, research has shown that trajectories of a computer cursor in a choice-reaching task reveal the performer’s uncertainly and ambivalence in perceptual and numerical judgment [
31,
32,
33,
34], linguistic judgment [
35], social categorization [
36], reasoning [
37], and economic choices [
38]. Mouse-cursor trajectories in choice reaching are also shown to reflect people’s emotion [
39] and attention deficit profiles [
40].
To investigate the impact of mindsets on target selection behavior, we employed a concept-learning task in which participants learned to classify probabilistically arranged geometric cards by trial and error (150 trials). In this task, participants had to move the cursor and select to click one of the two buttons to respond. In each trial, we tracked the movement of the computer cursor every 20 milliseconds and analyzed whether different motion patterns would emerge as a function of experimentally induced mindsets.
Our concept learning task was a modified version of the neuropsychological test developed by Knowlton and colleagues [
41]. Participants received 14 combinations of cards one at a time (150 trials in total) and learned to predict whether each combination belonged to “shine” or “rain” categories on the basis of feedback that was provided after each response (
Figure 1). Prior to the experiment, no information about card combinations and their outcomes (“shine” or “rain”) was given; thus, participants had to learn the concepts of “shine” or “rain” by trial and error. Each card was linked to the outcome of “shine” approximately 75, 57, 43, and 25% of the time.
To start a trial, the participant first pressed the Next button, the cursor was then placed automatically at the center of the button, and the stimulus picture (card combination) was presented on the monitor (
Figure 1). To indicate a selection, participants pressed a target button (either the left or right button shown at the top left/right corner of the screen). Soon after pressing the target button, the stimulus disappeared and feedback was presented (e.g., “Yes. It’s shine”). This cycle was repeated 150 times. For the entire experiment, our program recorded the
x–
y coordinate location of the cursor every 20 milliseconds.
Manipulating mindsets. To manipulate participants’ mindsets, we experimentally induced participants to believe, temporally, that people’s ability is fixed or malleable. First, participants (
N = 255) were randomly assigned to one of two conditions—growth-mindset condition or fixed-mindset condition—and read and memorized one of two vignettes as a part of a mock “memory test” [
42]. One vignette (growth-mindset condition) described the human intelligence as a malleable quality, and training and experience modifies one’s ability. The other vignette (fixed-mindset condition) characterized one’s intelligence as a fixed quality: it is inherited from parents and largely determined by genes (
Appendix A). After this mock “memory” test, all participants performed the concept learning task for about 20 min.
1.3. Modeling Mindsets by Kalman Filter
The Kalman filter, which has been applied widely for navigation control and robotic motion planning systems [
43], offers an ideal tool to quantify choice-reaching behavior in human computer interaction. To reach a target by the hand, the sensorimotor system needs to know the final location of the target, the current location of the hand, and motor procedures (muscle activity) to reach the next step in real time. But, sensory feedback is necessarily delayed by conduction and transmission lags (a monosynaptic stretch reflect requires 40~80 ms delay) [
44]. To compensate this lag, the neural system employs a hybrid of feedback and feedforward controllers, just as the Kalman filter coordinates [
44,
45]. In neuroscience, this algorithm has been adopted to model action potentials in neurons (e.g., Hodgkin–Huxley model, and see [
44]). In the current study, a Kalman filter to model mindsets revealed in trajectories of target selection was applied.
A system model of a Kalman filter is shown in (1). Here,
xk+1 (state variable) represents the unknown position of the computer cursor at time
k + 1 and is linearly related to the previous state
xk by transition matrix
A with Gaussian white noise
wk ~
N (0,
Q), where
Q is the covariance matrix of
wk;
zk is a measurement/observation of the state variable
xk. Roughly,
zk corresponds to output from a sensory systems that track the cursor position;
zk is linearly related to state variable
xk by matrix
H and mired by noise
vk ~
N (0,
R). Thus,
Q corresponds to the degree of precision (i.e., 1/
Q) of the actuator, whereas
R corresponds to the degree of precision (i.e., 1/
R) of the sensory feedback system. In our formulation, the true state
xk is unknown, and is estimated (
) by sensory output
zk.
The computational algorithm of a Kalman filter is shown in
Figure 2. Here, the overhead notation “^” stands for an estimated value and “-“ represents a predicted value. For example, in (3) (
, the predicted estimate of the state at time
k (i.e.,
is obtained by multiplying the estimate of the state at
k − 1 (
) by transition matrix
A.
The algorithm starts with pre-specified initial values,
and
P0, which represent an initial estimate of the state (locations) of the system (
) and error covariance
P0, the degree of error of the initial estimate (Step 0 in
Figure 2). From here, the algorithm iteratively estimates state
(e.g.,
x–
y coordinate locations and velocity of the cursor) in each time step by coordinating sensory observation
zk and its estimate (
), which are weighted by
Kk (Kalman gain at time
k) (Step 3). Both Kalman gain
K and error covariance
Pk are computed iteratively in Steps 2 and 4. In this process, pre-specified system-specific error covariance
Q (Step 1) and observation-specific error covariance
R are internalized. The system receives observation (input or recorded cursor positions)
zk at each time step and corrects an estimate of the “true” unknown state
based on the forward estimate (
) and feedback (posterior) estimate weighted by Kalman gain
K (
).
In our formulation, we define state variable
x with a four-dimensional vector representing
x–
y coordinate locations (
ϒx,
ϒy) of the cursor and their velocity (
vx, vy) ([
ϒx,
vx, ϒy, vy]
T). Observation variable
z only has a
x–
y coordinate location; [input_
x, input_
y]
T and the velocity associated with the
x–
y coordinate is assumed to be unobservable for the sensory system. Furthermore, the transition matrix
A corresponds to a default motor plan that the system possesses. Following the empirical findings in [
46], I assume that two competing motor plans (
Aleft and
Aright) are simultaneously formed (
A = α
Aleft + (1 − α)
Aright).
Assuming that subjects have no a priori inclination to move to the left or right, I set α = 0.5 (
A = (
Aleft +
Aright)/2) (3). Thus, transition matrix
A is analogous to a constant motion model (p. 20, [
43]), in which
x–
y coordinate locations at state
xk+1 are estimated from
xk by adding a multiple of velocity and time increment
dt (we defined
dt = 20 ms) (e.g.,
xk+1 =
Axk +
wk). In our experiment, every cursor motion starts with the same starting position (
x,
y) = (0, 0) (10) and all other cursor locations are specified with respect to the original starting position. To capture individual differences in sensorimotor capacity of the participants, the initial value of error covariance
P0 was set randomly with Gaussian white noise (
P0 ~
N (mu = 40, sigma = 10) (11). For each participant, actuator error
Q is set randomly with Gaussian noise (
Q ~
N(10, 2)) (12) and feedback error
R is set randomly with Gaussian noise (
R ~
N(50, 5)) (13):
In our Kalman filter model analysis, A (14), H (18), Q (12), and R (13) were initially preset. For model fitting, I treated one of these variables as free parameters and sought appropriate values using an expectation maximization (EM) algorithm. For example, in one set of model fitting, we treated A, H, and Q as fixed (as specified in (14), (18), (12)) and R as free parameters and the values of R were specified by the expectation maximization (EM) algorithm. In another set of model fitting, we treated A, H, R as fixed and Q as free parameters, and so on.
The critical questions addressed in this study were to identify the extent to which (a) experimentally induced mindsets would influence mouse motion trajectories in target selection and (b) our Kalman filter models would effectively separate motion trajectory patterns elicited in the two mindset conditions. Specifically, we seek the extent to which system-specific error covariance Q (Equation (1)) and observation-specific error covariance R (Equation (2)) separate the two mindset conditions. Q and R correspond to variance (error) pertaining to the prediction and estimation of the state and observation (Equations (1) and (2)). Thus, when the trajectory is straightforward (e.g., little variability in x–y coordinate location), Q and R will be small. In contrast, if the trajectory is unpredictable/unstable and highly variable (e.g., zigzag), then Q and R will be large. Individual elements of Q and R matrices indicate specific dimensions of instability (variability). For example, the trajectory is unstable along the x location, then Q[1, 1] will be large. If the trajectory is unstable along the y location, then Q[3, 3] will be large. Relative magnitudes of Q and R (large/small Q or large/small R) are also associated with the size of K (Kalman gain, Equation (7)), which in turn determines relative weight given to system-based or observation-based prediction/estimation (Equations (1) and (2)).
Research shows that mindsets influence the level of task engagement [
6]. Those who have growth mindsets have an elevated level of task engagement while those with fixed mindsets tend to engage in the task less. We think that the level of engagement is reflected in the stability of cursor trajectories. Those who are more engaged show more stable and consistent trajectories, while those who are less engaged in the task show unstable and highly erratic trajectories. On this basis, we think that participants induced with growth mindsets will show more stable and ordinary trajectories (smaller
Q and
R), as compared to those induced with fixed mindsets.