Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation

The first stage: Combining the job responsibilities of all psychological teachers in mental health education colleges and universities and adopting the literature method and expert consultation method to determine the performance evaluation standards. Or all-member psychology teachers of provincial-level outstanding colleges and universities, outstanding educators, all-employee psychology teachers of model colleges and universities, advanced workers in the education system, etc. or all-member psychology teachers of colleges and universities who have been commended for outstanding achievements in mental health education at the national or provincial level 2) All college psychological teachers who have been engaged in mental health education for >4 years.

The second stage: According to the above evaluation criteria, 5 excellent performers and 5 ordinary performers are selected from all primary and secondary schools in Chongqing as interview subjects. The gender composition is 4 females in the excellent group and 1 male; and 4 females in the general group and 1 male. The school category is composed of 4 psychological teachers in the excellent group of middle schools and colleges, 1 psychological teacher in the elementary schools; and 4 psychological teachers in the general middle schools and colleges, and 1 psychological teacher in the primary schools. Then, the researchers conducted behavioural event interviews with the subjects, that is, the interviewees were asked to narrate their key cases in the mental health education work, including 3 successful events, unsuccessful events or negative events. The participants described in detail the cause, process, result, time, relevant people, the scope of the incident, level of influence and their own thoughts or feelings at the time and recorded the content of the interview [2].

The third stage: Organising the interview recordings into manuscripts and using Spencer's competency coding dictionary to encode the interview data. A coding team composed of two people respectively coded an interview recording manuscript. After the independent coding was completed, a comparative discussion was conducted to ensure that the two coders reached a consensus on the coding of the interview manuscript. After unifying the opinions, modify the competency rating scale to form a mental health education feature rating scale for all-staff mental health education activities, and then code and score all recorded texts, and use SPSS12.0 to analyse the data obtained by coding deal with.

Introducing the fractal space-time distance of the differential equation of one-dimensional space and time (1) {Δt^=ΔtaΔx^=Δxβ \left\{ {\matrix{ {\Delta \widehat t = \Delta {t^a}} \hfill \cr {\Delta \widehat x = \Delta {x^\beta }} \cr } } \right. where α is the fractal dimension of time and β is the fractal dimension of one-dimensional space. Obviously, the non-Euclidean distance defined by Eq. (1) is obtained based on two assumptions of fractal in deformability and fractal equivalence. Balankin et al. further gave the general three-dimensional fractal distance, and the differential equation fractal distance is a special case. (2) {Δta=ta−t0arβ=(xβ−xjβ)2+(yβ−yjβ)2+(zβ−zjβ)2 \left\{ {\matrix{ {\Delta {t^a} = {t^a} - t_0^a} \hfill \cr {{r^\beta } = \sqrt {{{({x^\beta } - x_j^\beta )}^2} + {{({y^\beta } - y_j^\beta )}^2} + {{({z^\beta } - z_j^\beta )}^2}} } \hfill \cr } } \right. where β is the fractal dimension of three-dimensional isotropic space. When β = 1, the fractal space distance of the differential equation of Eq. (2) reverts to the classic 3-dimensional Euclidean distance. If the initial time t₀ = 0 is set and the one-dimensional problem source point coordinate x_j = 0 is set, then the fractal distance definition Eq. (2) of the differential equation is simplified to Eq. (1). The derivative of the differential equation is defined as (3) {dudxβ=limx′→xu(x)−u(x′)xβ−xβ=1βxβ−1dudxdudta=limt′→tu(t)−u(t′)ta−t′a=1dta−1dudt \left\{ {\matrix{ {{{du} \over {d{x^\beta }}} = \mathop {{\rm{lim}}}\limits_{{x^\prime} \to x} {{u(x) - u({x^\prime})} \over {{x^\beta } - {x^\beta }}} = {1 \over {\beta {x^{\beta - 1}}}}{{du} \over {dx}}} \hfill \cr {{{du} \over {d{t^a}}} = \mathop {{\rm{lim}}}\limits_{{t^\prime} \to t} {{u(t) - u({t^\prime})} \over {{t^a} - {t^{'a}}}} = {1 \over {d{t^{a - 1}}}}{{du} \over {dt}}} \hfill \cr } } \right. The literature gives the basic solution of the differential equation diffusion process of a one-dimensional problem. The derivative diffusion equation of the three-dimensional differential equation is (4) ∂udta=D(∂udxβ(∂udxβ)+∂udyβ(∂udyβ)+∂udzβ(∂udzβ)) {{\partial u} \over {d{t^a}}} = D\left( {{{\partial u} \over {d{x^\beta }}}\left( {{{\partial u} \over {d{x^\beta }}}} \right) + {{\partial u} \over {d{y^\beta }}}\left( {{{\partial u} \over {d{y^\beta }}}} \right) + {{\partial u} \over {d{z^\beta }}}\left( {{{\partial u} \over {d{z^\beta }}}} \right)} \right) In the formula, u is the concentration of the diffusing substance and D is the diffusion coefficient. It can be easily proved that Eq. (5) is the basic solution of diffusion Eq. (4). (5) u(t,x,y,z)=H(t−τ)(aπDΔta)3/2exp(−r2β4DΔta) u(t,x,y,z) = {{H(t - \tau )} \over {{{(a\pi D\Delta {t^a})}^{3/2}}}}\exp \left( { - {{{r^{2\beta }}} \over {4D\Delta {t^a}}}} \right) In the formula, H is the multi-dimensional competition step function. The exponential term on the right side of Eq. (5) is the extended Gaussian distribution widely used in science and engineering. It can be seen from Eq. (5) that the basic solution of differential equation derivative diffusion Eq. (4) characterises the statistical diffusion process corresponding to the extended Gaussian distribution. The probability density function of the stretched Gaussian distribution is the kernel function of the basic solution of diffusion. When α = β = 1, the basic solution of the fractal distance of the differential equation of Eq. (5) returns to the basic solution of the classical integer-order Euclidean distance diffusion equation, describing the Gaussian distribution of particle motion in the process of classical Fick diffusion (normal diffusion) feature. Below, this article considers the diffusion equation containing only the derivative of the time differential equation (6) dudta=−Du {{du} \over {d{t^a}}} = - Du The solution of this equation is (7) u=Cexp(−DΔta) u = C\exp ( - D\Delta {t^a}) The constant C here is determined by the initial conditions. Equation (7) is the exponential decay of extension that often appears in the literature. When α = 1, the solution of the derivative model of the differential equation degenerates to the Debye exponential decay of the classical integer-order diffusion equation. It can be proved that the derivative Laplace equation, wave equation, Helmholtz equation, convection-diffusion equation, etc. of the differential equation also satisfy the basic solution of the fractal distance of the differential equation. In addition, substituting the structure function for the power function in the fractal distance definition of the differential Eqs. (1) and (2), the structure distance of the general fractal (structure shape) describing the non-power law function is obtained. (8) {ΔG(t)=G(t)−G(t0)Q(r)=(Q(x)−Q(xj))2+(Q(y)−(yj))2+(Q(z)−(zj))2 \left\{ {\matrix{ {\Delta G(t) = G(t) - G({t_0})} \hfill \cr {Q(r) = \sqrt {{{(Q(x) - Q({x_j}))}^2} + {{(Q(y) - ({y_j}))}^2} + {{(Q(z) - ({z_j}))}^2}} } \hfill \cr } } \right. where G and Q are the structure functions in the derivative of time and space structure respectively. The definition of fractal distance by Balankin et al. actually includes the definition of Eq. (8). The basic solution of the local structure derivative diffusion equation can be directly obtained by using the above structure distance.

Without loss of generality, consider a particle moving along a one-dimensional curve in fractal time at a constant velocity. The relationship between movement distance and time is (9) l(τ)=υ(τ−t0)a l(\tau ) = \upsilon {(\tau - {t_0})^a} where l is the distance, v is the uniform velocity, τ is the current time, t₀ is the initial moment and α is the time fractal dimension. If the speed is not uniform, the corresponding integral of the differential equation is (10) l(t)=∫t0tυ(τ)d(τ−t0)a l(t) = \int_{{t_0}}^t \upsilon (\tau )d{(\tau - {t_0})^a} From Eq. (10), the corresponding derivative expression of the differential equation can be obtained (11) dldta=limt′→tl(t)−l(t′)(t−t0)a−(t′−t0)a=1α(t−t0)a−1dldt {{dl} \over {d{t^a}}} = \mathop {\lim }\limits_{{t^\prime} \to t} {{l(t) - l({t^\prime})} \over {{{(t - {t_0})}^a} - {{({t^\prime} - {t_0})}^a}}} = {1 \over {\alpha {{(t - {t_0})}^{a - 1}}}}{{dl} \over {dt}} Comparing Eqs. (3) and (11), the two differential equation derivative definitions, notice that the only difference is that the latter includes the initial time while the former assumes that the initial time is 0, and so the latter is a more general expression.

The position of the particle motion expressed by Eq. (9) at time τ can also be calculated by Eq. (12). (12) l(τ)=υ(t−t0)a−υ(t−τ)a l(\tau ) = \upsilon {(t - {t_0})^a} - \upsilon {(t - \tau )^a} where t is the end time. The first term on the right side of Eq. (12) is the total movement distance of the particle, and the second term represents the distance to be moved from time τ to the end time t. If it is not a constant velocity motion and the first-order differential operation of the τ variable to Eq. (12), there is (13) dl=−υd(t−τ)a dl = - \upsilon d{(t - \tau )^a} The corresponding integral of Eq. (13) is expressed as (14) l(t)=∫t0t−υ(τ)d(t−τ)a l(t) = \int_{{t_0}}^t - \upsilon (\tau )d{(t - \tau )^a} Note that an essential difference between Eqs. (14) and (10) is that Eq. (14) is a Volterra integral with a power function kernel.

According to the above analysis, it can be seen that the differential equation calculus and fractional calculus are intrinsically quantitatively related to the fractal dimension, that is, the order of the time differential equation calculus and fractional calculus is the time fractal dimension α of the research object.

From the perspective of time fractal analysis, differential equation calculus and fractional calculus are opposite processes. The former advances gradually from the initial moment, while the latter recursively from the end moment. From the time history analysis, there is a certain reverse correspondence between the two [3].

On the other hand, differential equation calculus is a local operator and fractional calculus is a non-local operator. Therefore, even if both are used to describe the fractal process, the corresponding statistical process is completely different. They are two different calculus operators. In addition, the geometric basis of the derivative equations of differential equations is the non-Euclidean differential equation fractal distance, while the geometric basis of the fractional calculus is still the Euclidean distance. The statistical mechanics background of the two is also completely different. The differential equation derivative diffusion equation model describes the spreading Gaussian distribution and the spreading exponential attenuation diffusion, while the fractional derivative diffusion equation describes the Levatich steady-state distribution and the ML attenuation diffusion. The comparison of exponential decay, stretch exponential relaxation and ML decay is shown in Figure 1.

Fig. 1

The interview lengths of 10 interview texts were analysed and statistically compared. Among them, 5 were in the excellent group and 5 were in the ordinary group. The results are shown in Table 1.

In order to ensure that the differences in the competency characteristics between the excellent group and the ordinary group are not caused by the length of the interview, the number of words and interview time between the excellent group and the ordinary group was tested for significance. The results are shown in Table 2.

The average word count of the excellent group interview was 4035 words and the average time was 2304 s; and the average word count of the ordinary group interview was 3281 words and the average time was 2040 s. The difference between the excellent group and the ordinary group in terms of interview text and interview time was not significant at the 0.05 level. It can be seen that the number of words and the length of time did not affect the difference in competency between the excellent group and the ordinary group.

After discussing and analysing the pre-coded text, the two coders agreed that they can learn from interpersonal understanding and communication, respect for students, sense of responsibility, student service orientation, enthusiasm, relationship establishment, self-regulation and control, self-confidence, initiative, desire for achievement, flexibility, influence, honesty and integrity, promotion of student development, career preferences, organisational coordination, professional knowledge and skills, information collection, analytical thinking, reflective ability, challenge and support, innovation and adaptability. These are 23 competency characteristics. It depends on whether all the psychological teachers of the mental health education colleges and universities have these competent characteristics [4].

The two coders formally coded and scored the interview recordings according to the coding dictionary of mental health education activities for all employees. The coding and scoring were performed independently, back to back. The degree of consistency of independent coding results for the same text is an important factor affecting the reliability of the competency evaluation method, and an important indicator of the reliability and objectivity of the coding results. In order to test whether the scores between the two coders are consistent, this study conducted the scorer reliability test which is shown in Table 3.

In Hay's classic research, three indicators, frequency, average grade score, and highest score, are used for statistical analysis. Among them, the average grade score is the best. Research by Shikan and others in China has also confirmed this result. This study mainly used two coders to test the average grade score of the text. The test results show that the two coders have a high degree of consistency in each competency, with an overall consistency of 0.979, which is higher than that in previous studies.

Using the average grade score as an indicator, the average scores of the two coders on each competency feature are calculated and then the average scores of the excellent group and the ordinary group on each competency feature to test the significance of the difference are compared. The results are shown in Table 4.

From the test results, it can be seen that the excellent group and the ordinary group have extremely significant differences in interpersonal understanding and communication, desire for achievement, influence and reflective ability. They are in respect of students, student service orientation, self-control and adjustment, self-confidence and there are significant differences in the promotion of student development, organisation and coordination, professional knowledge and skills, and analytical thinking; but there are no significant differences between the two groups in other competency characteristics. Therefore, the 12 competence characteristics of all psychological teachers in mental health education colleges are interpersonal understanding and communication, respect for students, student service orientation, self-regulation and control, self-confidence, desire for achievement, influence, promotion of student development, organisation and coordination, professional knowledge skills, analytical thinking and reflective ability. The test results also show that the average grade scores of excellent and ordinary people are not high in the competence of analytical thinking. Researchers believe that, on the one hand, it may be caused by the coder's scoring standards. On the other hand, in the key events described, the descriptions of outstanding performers and ordinary performers did not show the higher-level behaviours in the competency coding dictionary characteristics, and so the score is not high [5].

According to the test results of the difference of interview data, find out the competencies with significant differences, and determine the competencies of all psychological teachers in mental health education colleges as: interpersonal understanding and communication, respect for students, student service orientation, self-regulation and control, self-confidence 12 competency characteristics: heart, desire for achievement, influence, promotion of student development, organisation and coordination, professional knowledge and skills, analytical thinking and reflective ability. According to Spencer's classification of competency characteristics, many competency characteristics contained in the mental health education activity characteristic model of mental health education for all employees are divided into four competency characteristics dimensions. See Table 5.

The total frequency of the mental health education activities of all the mental health education activities mentioned by the 10 subjects during the interview was ranked from high to low. The results are shown in Table 6.

It can be seen from Table 6 that among the 12 features of the mental health education activity feature model for all employees of mental health education, interpersonal understanding and communication, self-regulation and control, respect for students, student service orientation, and promotion of student development appear more frequently than other features, which shows that for all psychological teachers in mental health education colleges and universities, these 5 characteristics are more important than other characteristics.