**6. Dynamics**

52 Behavior Analysis

Process:

obtain μ

r = n − 1 = 11–1 = 10

*t*

*t n*−1;1 1−\_\_ *α* 2 = *t*10−1;1−\_\_\_\_ 0.05 2 = *t*

H0: μ = 0.56, the instrument is calibrated for accuracy.

*<sup>t</sup>* <sup>=</sup> *<sup>Y</sup>*¯ <sup>−</sup> *<sup>μ</sup>* \_\_\_\_

*n*−1;;1−\_\_ *α* 2 = { *t*

**Graph 2.** Normal distribution of the indicators t.

The numerical values of t extracted from [34].

with <sup>Y</sup>¯ <sup>=</sup> 0.63, *<sup>S</sup>* <sup>=</sup> 0.322, *<sup>α</sup>* at level 0.05, *<sup>r</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>11</sup> <sup>−</sup> <sup>1</sup> <sup>=</sup> <sup>10</sup> <sup>=</sup> degrees of freedom where <sup>n</sup> is the number of measurements.

confidence interval <sup>=</sup> 0.63 <sup>±</sup> 2.228<sup>∗</sup> \_\_\_\_\_ 0.322

= \_\_\_\_\_\_\_\_ 0.63 <sup>−</sup> 0.56 \_\_\_\_\_ 0.322 √ \_\_\_ 11

Different observational monitoring teams were employed to perform the measurements and

10;0.99 = 2.764;  *t*

Finally, we want to know if the measuring instrument calibrated for accuracy.

H1: μ ≠ 0.56, the instrument is not calibrated. There is a systematic error.

\_\_*S* √ \_\_ *n*

where *Y*¯ = 0.63, μ = 0.56 is a control team (calibrated with valid procedures).

10;0.95 = 1.812;  *t*

**Graph 2** represents the Normal Distribution of the indicators t.

The accuracy calibrated instrument hypothesis is accepted.

**Hypothesis test**: defined H<sup>0</sup> = Null Hypothesis and H<sup>1</sup> = Alternative hypothesis [35].

√ \_\_\_ <sup>11</sup> <sup>=</sup> {

9;0.975 = 2.228 ( for *Distribution t*‐Student).

0.41 (minimum) 0.85 (maximum)

≈ 0.72 (4)

10;0.999 = 3.169

(3)

From the time series of X (t), Y (t), and Z (t), the discretized column vectors are constructed (observed that although vectors with a minimum of 1000 elements allow making good estimations, the ideal is that contain over 5000 components for the stability of the Lyapunov coefficients). According to the significant learning of the team of students, the graphs of the time series in the phased space acquire such forms:

The time series and the graph obtained are those that allow incorporating elements of chaos theory in their study. They satisfy two fundamental conditions of this theory: sensitivity to initial conditions and the existence of Lyapunov exponents greater than zero. Applying to the experimental data, the Lorenz equations [36, 37] modified according to the fourth order Runge Kutta numerical method, the dynamics classified from the control parameter

(also called connectivity). Its values deliver the performance of the teams that make up the Experimental Group: Low (r = 16.5, weak attractor), Medium (r = 20.5, medium attractor), and High (r = 28.7, chaotic attractor).

**7. Connectivity**

dynamics:

study?

characterize them, in a first approximation.

**Graph 3.** Dynamics of learning versus connectivity.

The control parameter r (connectivity [40]) gives the transition between the different dynamics that favor meaningful learning. Connectivity defined as the capacity shown by the components of a system to expand the actions of others by their actions and to expand their actions from the actions of others [41, 42]. This definition is a glimpse into an underlying referential framework, inherent in all things, sustained by the complex intervariable interferences that

Initial Condition and Behavior Patterns in Learning Dynamics: Study of Complexity and…

http://dx.doi.org/10.5772/intechopen.74140

55

These interferences induce clutter dynamics that create an intelligent collective order, but temporary, which makes it imperative to incorporate them in learning. Teams with high connectivity and high POS/NEG quotients (greater than or equal to: 2.5 [43], 4.3 [44], and 5 [17]) are sustained over time and achieve the objectives of the activity [24, 45]. When observing **Graph 3**, we can see a growth in connectivity, as we approach the chaotic or complex

What does this increasing behavior of connectivity (entropic connectivity) mean for learning? Is it possible to calculate it? How is it related to the complexity of the learning process under

Answering these questions, different numerical procedures were applied to the time series [46], which allow determining the Lyapunov coefficients [47], the Kolmogorov entropy (SK)

The position of Ilya Prigogine [52, 53] on irreversibility and entropy varies that of traditional

"Entropy always contains two dialectical elements: a creator element of disorder, but also a creative element of order. (…) We see, then, that instability, fluctuations, and irreversibility

[48, 49], the complexity [50], and finally, the uncertainty in information [51].

physics. In his lecture The Birth of Time (Rome, 1987), Prigogine argued:

**8. Irreversibility and the sustainability of learning**

These values compared with those that arise from theoretical iterative cycles (using, for example, adjustment by Fourier Time Series) for X (t), Y (t), and Z (t), based on the range of their experimental domains. Programming in MatLab (software for numerical calculation and scientific analysis) the modified Lorenz equations [28, 33] (also possible by Neural Networks [38] or Cellular Automata [39]), the graphs are obtained as shown:

These graphics are classified according to the values of the theoretic control parameter, r, which roughly matches with the values of r for weak, medium and chaotic attractor, respectively, emerging from the experimental Time Series.

It was observed that the contrast between the performance of the experimental groups (selecting a team with chaotic dynamics) and the control groups (traditional courses without initial condition: choosing a good performance team) is carried out through cross-correlation. The cross-correlations by group according to the influence exerted by the variable of emotions Y (= Positivity/Negativity) on the variable X (= Inquiry/Persuasion) [20–25] is observed in **Table 2**.

The experimental group treated with contextualized initial conditions, which promote high connectivity within each team, shows that the balanced presence of positivity/negativity in their relationships exerts an influence 1.7 ~ 2, approximately, on the variable X, which is inquiry/ persuasion (the most rational part of the team's work). Thus, the team leads more efficiently and safely toward the achievement of meaningful learning. This influence translated into connectivity and emotional field evolution reflected in the value that students give to learning and in its achievements. These achievements range from the experience of collaborative work, each component is determined in the learning process, to the formal evaluation procedures applied ranging from the weekly reports, entrance test at the beginning of the teaching session, tests, oral interrogation of any component of the team whose performance is extended to the whole team, etc.


**Table 2.** Comparison between cross-correlation according to the Control and Experimental Teams.
