*Article* **Measurement-While-Drilling Based Estimation of Dynamic Penetrometer Values Using Decision Trees and Random Forests**

**Eduardo Martínez García 1,2, Marcos García Alberti 1,\* and Antonio Alfonso Arcos Álvarez 3**

**Abstract:** Machine learning is a branch of artificial intelligence (AI) that consists of the application of various algorithms to obtain information from large data sets. These algorithms are especially useful to solve nonlinear problems that appear frequently in some engineering fields. Geotechnical engineering presents situations with complex relationships of multiple variables, making it an ideal field for the application of machine learning techniques. Thus, these techniques have already been applied with a certain degree of success to determine such things as soil parameters, admissible load, settlement, or slope stability. Moreover, dynamic penetrometers are a very common type of test in geotechnical studies, and, in many cases, they are used to design the foundation solution. In addition, its continuous nature allows us to know the variations of the terrain profile. The objective of this study was to correlate the drilling parameters of deep foundation machinery (Measurement-While-Drilling, MWD) with the number of blows of the dynamic penetrometer test. Therefore, the drilling logs could be equated with said tests, providing information that can be easily interpreted by a geotechnical engineer and that would allow the validation of the design hypotheses. Decision trees and random forest algorithms have been used for this purpose. The ability of these algorithms to replicate the complex relationships between drilling parameters and terrain characteristics has allowed obtaining a reliable reproduction of the penetrometric profile of the traversed soil.

**Keywords:** machine learning; decision trees; random forests; penetrometer; MWD; rigid inclusions

#### **1. Introduction**

Machine learning (ML) is a field in great health. As a concept, it is not something new (the first decision tree algorithm dates from 1963 [1]), but the current capacity to acquire and handle a huge amount of data has exponentially increased its popularity as an analysis tool. ML is a branch of artificial intelligence that consists of the application, through computer programs, of a series of algorithms to obtain information from large data sets. The term learning is used given that these systems are capable of improving and adapting to the information supplied to them.

Perhaps some of the best-known applications are those related to advertising and economics. For example, based on our browsing data, the algorithms can predict our preferences and offer us personalized advertising [2]. ML has also been widely used in medicine, where algorithms work as a medical diagnosis. As a mode of example, Scikitlearn, a machine learning library for Python, comes preloaded with small data sets on diabetes, exercise, and breast cancer [3]. In areas such as the industrial sector, the main aim of ML has been to predict when breakdowns will occur [4]. Mining, a field with problems similar to those of civil and geotechnical engineering, has also applied machine learning techniques [5].

**Citation:** García, E.M.; Alberti, M.G.; Arcos Álvarez, A.A.

Measurement-While-Drilling Based Estimation of Dynamic Penetrometer Values Using Decision Trees and Random Forests. *Appl. Sci.* **2022**, *12*, 4565. https://doi.org/10.3390/ app12094565

Academic Editors: Małgorzata Jastrz ˛ebska, Krystyna Kazimierowicz-Frankowska, Gabriele Chiaro and Jaroslaw Rybak

Received: 9 March 2022 Accepted: 27 April 2022 Published: 30 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In civil engineering, there have already been important contributions to the monitoring of dams [6], excavations [7], and traffic prediction [8]. However, there is still a lack of studies considering the large amount of data and ML in certain fields such as geotechnics.

Geotechnical engineering is a field whose problems present complex, non-linear relationships among its variables. These problems have been solved using empirical formulations that approximate the solution with more or less accuracy. ML algorithms have shown to be extremely efficient to analyze non-linear problems. Therefore, they are an ideal tool to be applied to geotechnics. One of the obvious applications is to predict the behavior of the soil from parameters obtained from laboratory tests. This approach achieves higher degrees of accuracy than the traditional analytical formulas [9]. However, the application of ML has not only been limited to geotechnical parameters but also to computing load-bearing capacity, settlement, liquefaction and slope stability [10–12].

Modern geotechnical machinery and equipment has abundant instrumentation that allows the attainment of a multitude of drilling parameters such as the drilling speed or the applied torque. This constant collection of data is usually named Measurement-While-Drilling (MWD). This record supplies an interesting source of information for the application of machine learning techniques.

On another note, continuous dynamic penetration tests consist of the introduction of a penetration element into the ground by striking a hammer with a defined weight. The result of the test is the number of blows required to advance a certain length. This type of test has become one of the most attractive alternatives giving several advantages:

The data obtained from MWD is in some way related to the characteristics of the terrain. These data are recorded continuously throughout the perforation and should be able to be correlated with the blows of the dynamic penetrometer. However, this relationship is unknown, complex, and non-linear.

In this study, a significant amount of data obtained from the MWD system from deep foundation and ground improvement machinery is analyzed. Figure 1 shows the plan views of the four sites studied. The circles represent the perforations made; the crosses are the penetrometers available. The aim was to correlate said penetration tests with the values obtained from MWD using ML techniques, specifically decision trees and random forests.

The significance of this research lies in the fact that, at the time of writing, it is the first time that this approach has been applied to rigid inclusions execution data.

The input variables have been the drilling speed (m/h), rotation speed (rpm), rotation torque (t·m) and thrust (t). The objective variable has been the blow values of dynamic penetrometers.

The results obtained show that the algorithms have been able to relate, with a certain degree of precision, the drilling parameters with the blows of the penetrometers. This information allows an increase in the level of security, an adaption of the execution to the real conditions of the ground and detection of possible anomalies, among other things.

The structure of the paper is the following:

**Figure 1.** Distribution of columns (circles) and penetrometers (crosses) available in the analyzed sites. (**a**) Site 1. (**b**) Site 2. (**c**) Site 3. (**d**) Site 4.

#### **2. Related Works**

Geotechnical engineering hosts problems with many uncertainties in which the true relationships among the variables are unknown. This makes it an interesting field for the application of ML algorithms. In this way, several authors have used this approach [10], especially through the application of neural networks (NN).

In the field of layer characterization, Shuku et al. [13] proposed a new method for estimating trends and layer boundaries in depth-dependent soil data based on lasso. This new method (SBLasso) was applied to synthetic data, as well as an actual CPT sounding taken at Texas A&M University and it provided stratification consistent with existing methods.

Zhao and Wang [14] proposed an interpolation first method to characterize a multilayer soil property profile when measurements within each layer are sparse and limited. This method interpolates the multilayer soil property profile using measurements from all layers together as input to a Bayesian supervised machine learning first. Then, the interpolated multilayer soil property profile is stratified using an unsupervised machine learning method, e.g., the modified k-means clustering method.

Focusing on the use of MWD, Rai et al. [15] reviewed MWD techniques in the extractive industry, and therefore focused on rock drilling. Kadkhodaie-Ilkhchi et al. [16] conducted a comparative study of three machine learning techniques using MWD data from drilling for explosives at an iron mine in Australia. Other studies related to the recognition of rocks using MWD and ML have also been published with such references [17–20].

In the field of soils and civil engineering, Goh [21,22] estimated the load-bearing capacity of driven piles using NN in non-cohesive soils from data collected by Flaate [23] for wooden, precast concrete and steel piles. The input parameters were hammer weight, hammer drop height, hammer type, pile length, pile weight, pile modulus of elasticity, pile cross-sectional area, and pile set. The target value was the load-bearing capacity of the pile. The model was able to obtain high correlation coefficients between the input parameters and the load-bearing capacity for both the train set and the test set. Furthermore, NN were found to perform better than classical formulas when predicting load-bearing capacity.

Lee and Lee [24] also used NN to calculate the load-bearing capacity of piles. Five input variables were used: penetration depth ratio, the average SPT strike along the axis of the pile, the average SPT strike near the tip of the pile, the penetration rate for each strike and the hammer energy. The results obtained were compared with the Meyerhof formula [25]. The values obtained through NN correlated better with the measured values than those obtained through the Meyerhof equation.

Although the input parameters did not come from MWD, Teh et al. [26], proposed an NN model to estimate the load-bearing capacity of piles from dynamic pressure wave data. The data came from 37 precast piles from 21 different sites. The objective parameters were the values of the soil parameters predicted by the CAPWAP model [27]. This approach to predicting parameter values is similar to what is considered in this article.

In another use of NN, Diaz et al. [28] predicted the rate of penetration during wellbore drilling using data obtained as the drilling progresses within a given well.

Pal and Deswal [29] used a Gaussian regression process (GP) to predict the loadbearing capacity of the pile. Part of the input data was the same as that used by Goh. The performance of the proposed model was compared with support vector machines (SVM) and empirical relationships, obtaining a better result.

Galende-Hernández et al. [30] carried out a study estimating the value of the RMR from the characterization of the excavation face of a tunnel, using MWD and expert knowledge in the execution of a tunnel using explosives. The results obtained showed a good correlation. Such work posed a similar philosophy to that of this article since it sought to estimate the value of a soil parameter (the RMR) from the drilling parameters.

Although several examples of the use of MWD and ML data have been collected, the use of data obtained directly from machinery is relatively scarce. In this proposal, data from construction sites was taken directly from the MWD system and used to estimate the values of the dynamic penetrometer. The objective was to reach results for each perforation to be equivalent to having carried out a penetration test. Although data from rigid inclusions with ground displacement has been used, the methodology is general enough to be applicable to other works.

#### **3. MWD in Pile Driving and Soil Improvement Machinery**

The current pile machinery has a series of measuring instruments that allow the collection of performance data. The main information obtained during rotary drilling usually consists of the following [31]: the length of the drilling, drilling speed (*VA*), rotation speed (*VR*) and the hydraulic pressures of the rotary and thrust motors. These data are obtained with a high frequency, for example, every 10 cm of perforation.

#### *3.1. Direct and Indirect Parameters*

The parameters obtained from MWD can be divided into two groups depending on the process of attainment: direct parameters and indirect parameters.

Directly obtained parameters are those in which it is not necessary to apply any type of correlation to the values obtained from the sensors. Depth, rotational speed, and drilling speed can be considered direct parameters as they are obtained directly or as a function of time. Perhaps the most relevant of this type of parameter is the drilling speed as it is very indicative of the hardness of the terrain.

However, other parameters such as thrust force (PO) and rotation torque (CR) are obtained through mechanical correlations from the hydraulic pressure of the motors. These values (although they can be provided by the manufacturer) depend on a multitude of factors: the efficiency of the motor, the configuration of the machinery at any given time, the wear of the components, and so forth. Consequently, it is exceedingly difficult to know the true value of these parameters. These possible sources of error must be considered when analyzing the results obtained when using parameters that do not come from direct measurements. To this type of error attributed to the correlations, the errors and tolerances of the measuring devices must also be added.

#### *3.2. Compound Indices and Parameters*

Drilling parameters are related to soil resistance. However, by taking these parameters separately, the relationship is not clear. For this reason, various authors have developed compound parameters that combine individual parameters into energy expressions or empirical indices seeking to show the strength of the ground from the drilling data.

Most of these parameters or compound indices follow the same basic structure. In this way, the hardness of the ground is directly proportional to the rotation torque and the applied load and inversely proportional to the area of the drilling and drilling speed. These relationships tend to soften the profile, giving them greater physical meaning and making their interpretation easier. The most common of these indices are listed in Table 1 [32].

**Table 1.** Most used compound parameters.

Of these parameters, penetration resistance, Somerton's index, and drilling specific energy are intended for research in hard rock and soils or for recording parameters during on-site testing. Their application to soft soils, based on MWD data, can be difficult to interpret.

The main obstacle with these formulas is that they assume a form of the relationship between the parameters and the strength that, although logical, does not necessarily correspond to reality. Therefore, these formulas give us a qualitative idea of the resistance to perforation and are useful when the differences in behavior within the soil are clear, but they do not supply precise figures.

#### **4. Decision Trees and Random Forests**

#### *4.1. Decision Trees*

Decision trees (DT) are a popular algorithm for classification or regression capable of providing reliable and easily interpretable results.

Although there are a variety of DT algorithms, they all have a similar structure. To be concise, the algorithm consists of dividing the data set into successive subsets following

established rules. This process can be represented graphically in structures that resemble a tree (Figure 2).

**Figure 2.** Decision tree example for site 1, limited to 4 leaf nodes.

The success of DTs is explained by several factors that make them quite useful in practice [39]:

#### *4.2. Division Rules*

A division at a node can be understood as a question whose answer divides the set at the node into two or more subsets. These questions or division rules vary depending on the type of decision tree chosen. Once these rules have been applied to each of the variables in the set, they are ordered according to the information they provide, establishing the most useful division as a node.

The application of these criteria defines the shape of the tree.

#### *4.3. Stop Criteria*

Overfitting refers to the situation where the model fits the training data too much, leading to larger errors in the test data. To avoid this problem stopping criteria can be set. This set of rules can be regulated to achieve trees that are neither too short (too general) nor too long (overfitting problem).

The tree will stop its development by itself in two cases:

In the previous cases, it is not possible to gain purity in the nodes by continuing the divisions, so the algorithm stops. These nodes that are not divided are called leaf nodes. In addition to these situations, other criteria can be added. The most common approaches are:

• Set a node as a leaf node if it contains less than a minimum number of samples.

All these criteria must be defined by the user looking for the most adequate balance. Such balance can be difficult to achieve, which is why specially dedicated models are commonly used at the expense of a greater computational load.

These stopping criteria can be understood as a method of pruning the tree. Specifically, they are pre-pruning methods that are carried out during the growth of the tree.

Another method would be to develop the entire tree and perform the pruning eliminating those nodes that provide the worst results in a different data set than that used to generate the tree. This method usually provides better results than pre-pruning.

#### *4.4. Assignment Rules*

The assignment rules refer to the criteria followed to assign a value to a leaf node. In this way, in regression problems, it would be assigned to the value that provides the smallest mean square error. In a classification problem, the value would be that of the most probable category.

#### *4.5. Implementation of DT in Scikit-Learn*

For the application of DT and RF, the Scikit-learn library has been used. Scikitlearn [40] is a Python module that integrates a wide range of machine learning algorithms. It is a package designed for non-experts, focused on the ease of use, documentation, and consistency of the application programming interface (API).

The learning is carried out using the fit method together with the train values (in the case of supervised learning, these values would be X\_train and y\_train tables for input variables and target variables to predict, respectively). During the definition of the model, the hyperparameters that control the algorithm are introduced. The following pseudocode shows an example of DT training where the minimum number of samples to divide the node is five.

```


*Article* **Effect of Composite Chitosan/Sodium Alginate Gel Coatings on the Quality of Fresh-Cut Purple-Flesh Sweet Potato**

**Chit-Swe Chit 1,†, Ibukunoluwa Fola Olawuyi 1,† , Jong Jin Park 1,2 and Won Young Lee 1,3,\***

† These authors are considered the first author.

**Abstract:** In this study, single-layer coating using chitosan (Ch) and sodium alginate (SA) solutions and their gel coating (ChCSA) formed by layer-by-layer (LbL) electrostatic deposition using calcium chloride (C) as a cross linking agent were prepared to improve storage qualities and shelf-life of freshcut purple-flesh sweet potatoes (PFSP). The preservative effects of single-layer coating in comparison with LbL on the quality parameters of fresh-cut PFSP, including color change, weight loss, firmness, microbial analysis, CO2 production, pH, solid content, total anthocyanin content (TAC), and total phenolic content (TPC) were evaluated during 16 days of storage at 5 ◦C. Uncoated samples were applicable as a control. The result established the effectiveness of coating in reducing microbial proliferation (~2 times), color changes (~3 times), and weight loss (~4 times) with negligible firmness losses after the storage period. In addition, TAC and TPC were better retained in the coated samples than in the uncoated samples. In contrast, quality deterioration was observed in the uncoated fresh cuts, which progressed with storage time. Relatively, gel-coating ChCSA showed superior effects in preserving the quality of fresh-cut PFSP and could be suggested as a commercial method for preserving fresh-cut purple-flesh sweet potato and other similar roots.

**Keywords:** purple-flesh sweet-potato; chitosan; sodium alginate; gel coating; preservation


*3.1. Failure Modes and Crack Propagation*

The failure modes of the FITRC and CRC columns under large eccentric loads were similar. The failure of all specimens was manifested by longitudinal bar yielding in the tension zone, followed by concrete crushing in the compression zone (Figure 8). The eccentricity of the FITRC45 and CRC45 columns was 0.60 and that of the FITRC35 and CRC35 columns was 0.70. In a previous study [55], the eccentric compressive behaviour of iron tailings sand RC columns was investigated, and similar results were obtained.

Cracks occurred in the tension zone at the midspan point along the lateral depth of the FITRC45 and CRC45 columns when the load reached approximately 0.2 *Nu* and propagated with increasing load, resulting in new microcracks. When the load reached approximately 0.9 *Nu*, vertical cracks were observed in the lateral compression zone of the column specimens and began to propagate. When the ultimate load *Nu* was reached, the concrete cracks in the compression zone elongated and were accompanied by the appearance of new microcracks. Some of the concrete was crushed as the test specimen continued to be loaded, and the ultimate compressive strain of the concrete in the compression zone was reached; thus, the axial load on the column specimens decreased. Similar failure processes were observed for FITRC35 and CRC35 columns.

Regardless of whether the eccentricity was 0.60 or 0.70, the final failure of all the RC columns was demonstrated by the crushed concrete in the outer layer of the stirrups and the intact concrete in the inner layer of the stirrups. This can be attributed to the fact that the stirrups significantly restrained the concrete in the inner layer and inhibited its deformation. In addition, Figure 8 shows that the area of crushed concrete was related to the eccentricity of the specimen section; that is, a smaller eccentricity indicated a larger area of crushed concrete.

Figure 9 shows the load–maximum crack width curves of the RC columns. The crack width propagation trends of FITRC and CRC columns remained similar for eccentricities of 0.60 and 0.70, and the curves of the four RC columns overlapped. Considering that the splitting tensile strength of FITC was lower than that of CC, the crack width of FITRC columns was larger than that of CRC columns at the same load level.

**Figure 9.** Load–maximum crack width curves of RC columns: (**a**) *e*0/*h*0 = 0.60; (**b**) *e*0/*h*0 = 0.70.

#### *3.2. Load–Deflection Relationships*

Figure 10 shows the axial load–deflection relationship (axial and lateral displacements) of RC columns under large eccentric loads. Axial and lateral displacements were measured using a vertical displacement gauge and a displacement gauge at the midspan point along the column depth, respectively. The negative and positive values in the figure indicate the axial and lateral displacements, respectively, at the midspan point along the column depth. Prior to longitudinal bar yielding, there was an approximately linear relationship between the loads and displacements of all the RC columns. After tensile yielding of the longitudinal bars, the stiffness of the RC columns decreased and the load-deflection curve increased nonlinearly until the peak load was reached when the concrete cover was crushed and spalled. After the peak load, the curve shows a decreasing trend; the load of the RC columns showed a rapid reduction of 5–10% followed by a more gradual reduction, and ductile failures occurred in the specimens. The vertical and lateral displacements of the CRC columns were clearly smaller than those of the FITRC columns. In particular, there were smaller lateral displacements compared to vertical displacements, considering that the modulus of elasticity of CC was approximately 19% greater than that of FITC (Table 6). Similarly, the peak loads of the CRC columns were slightly higher than those of the FITRC columns (Table 8).

**Figure 10.** Load–deflection curves of RC columns: (**a**) *e*0/*h*0 = 0.60; (**b**) *e*0/*h*0 = 0.70.

**Table 8.** Main performance indices of RC columns.

Figure 11 shows the lateral displacements of four typical RC columns distributed along the column depth (measured using five displacement gauges). The RC columns exhibited similar lateral displacements at the different loading stages. Furthermore, the columns showed slow and rapid increases in lateral displacement in the early loading stages and as the peak load approached, respectively. The lateral deformations of the RC columns were caused by first- and second-order moments. In accordance with the literature [56], the lateral deformations and sinusoidal waveforms of columns hinged at both ends remained similar. Therefore, the lateral deformation of the RC columns can be expressed as follows:

$$D\_L = \Delta\_p \sin\left(\frac{\pi \cdot l}{L}\right) \tag{1}$$

where Δ*p* is the maximum displacement at the midspan point along the column depth, *l* is the location along the column depth, *L* is the column depth, and *DL* is the lateral displacement of the column at l. Figure 11 compares the lateral deformations of the RC columns and the sinusoidal model, which agree well for eccentricities of both 0.60 and 0.70, thereby indicating that the sinusoidal model can effectively predict the lateral deformations of FITRC columns at different loading stages.

#### *3.3. Deformation and Ductility*

Table 8 lists the primary test results of the RC columns, including the peak load *Nu*, displacement at the corresponding peak load Δ*p*, and yield displacement Δ*y*. The behaviour parameters of the RC columns were quantified to determine the deformation capacity of the RC columns. In seismic design, the inelastic deformation capacity of specimens is generally

quantified using the displacement ductility factor [57] and deformation coefficient [58]. The displacement ductility factor indicates the ductility behaviour of the specimens, which can be calculated as follows.

$$
\mu = \frac{\Delta\_{0.85}}{\Delta\_y} \tag{2}
$$

The deformation coefficient indicates the deformation capacity of the specimens after reaching the peak load and can be calculated as follows.

$$
\lambda = \frac{\Delta\_{0.85}}{\Delta\_p} \tag{3}
$$

**Figure 11.** Lateral deformation along the section depth of typical RC columns: (**a**) FITRC45-2 column specimen; (**b**) CRC45 column specimen; (**c**) FITRC35-2 column specimen; (**d**) CRC35 column specimen.

Here, Δ0.85 is the corresponding displacement at *Nu* of 0.85 in the load decreasing stage [57], Δ*p* is the peak lateral deflection at the peak load, and Δ*y* is the corresponding displacement at the yield load when the limit is reached in the elastic stage. The yield displacement Δ*y* can be obtained using a graphical method [57] (Figure 12).

**Figure 12.** Definitions of yield displacement Δ*y* and ultimate displacement Δ0.85.

Table 8 shows that the mean ductility factors of the FITRC45 and FITRC35 columns were 1.78 and 2.82, respectively, and the ductility factors relative to the CRC45 and CRC35 columns were reduced by 36.6% and 19.5%, respectively. Therefore, an increase in section size and eccentricity reduced the difference in ductility between FITRC35 and CRC35 columns. A previous study [52] examined the axial compressive properties of full iron tailings RC columns and also found that the ductility coefficients of FITRC columns were lower than those of CRC columns. The mean deformation coefficients of the FITRC45 and FITRC35 columns were 1.33 and 1.35, respectively, and the deformation coefficients relative to CRC45 and CRC35 columns were reduced and increased by 5.27% and 2.80%, respectively. In the load-decreasing phase, the deformation capacities of FITRC and CRC columns were identical.

As can be seen from Table 8, the lateral deflection Δ*p* corresponding to the peak load of the FITRC columns was greater than that corresponding to the peak load of the CRC columns, and the mean values of Δ*p* of the FITRC45 and FITRC35 columns increased by 25% and 21%, respectively, compared to the CRC45 and CRC35 columns.

#### *3.4. Load–Strain Relationships*

Figure 13 shows the load–rebar strain curves of typical RC columns, where *c* is *Asc* of the compressive rebars, which were close to the axial compressive force with a negative strain value, and *t* is *Ast* of the tensile rebars, which were away from the axial compressive force (Figure 7) with a positive strain value. ε*y*14 and *εy*16 are the tensile yield strains of rebars with diameters of 14 and 16 mm, respectively. Figure 13 shows that the strains of *Ast* in the RC columns were greater than ε*y*14 and *εy*16, and the yield strain of rebars in the compression zone was reached before the peak load was reached. The stress state is a typical feature of compression failure at large eccentricities. In addition, the strain of rebars in the FITRC columns was significantly higher than that in the CRC columns for both *Asc* and *Ast*, because FITC had a lower modulus of elasticity than CC. In addition, the prismatic compressive and cracking strengths of FITC were slightly lower than those of CC. As a result, the ultimate load of the FITC columns was lower than that of the CRC columns, while the lateral deflection and axial displacement of the FITC columns are greater than those of the CRC columns.

Concrete strain gauges were placed along the section depth of the RC column, as shown in Figure 7a, to study the strain distribution of the concrete during loading. Figure 14 shows the strain distribution of typical concrete. Under large eccentric loads, the concrete in the tension zone of the RC columns cracked, resulting in failure of the concrete strain gauges near the tension zone. In particular, the strain distributions in the concrete before and after cracking were recorded. Figure 13 shows that the concrete strain was linearly distributed along the depth of the RC column section. Therefore, FITRC columns satisfied the planar section assumption, and the flexural capacity of FITRC columns can be theoretically calculated according to the planar section assumption.

**Figure 13.** Load–rebar strain curves of typical RC columns: (**a**) *e*0/*h*0 = 0.60; (**b**) *e*0/*h*0 = 0.70.

**Figure 14.** Typical concrete strain distribution along the section depth at the mid-span point: (**a**) FITRC45-2 column specimen; (**b**) CRC45 column specimen; (**c**) FITRC35-2 column specimen; (**d**) CRC35 column specimen.

#### **4. Analysis of Sectional Capacities**

#### *4.1. Moment Magnification Factor*

Considering the axial and flexural deformations that occurred in the RC columns under large eccentric loads during the experiment, the axial capacity, moment, and crack resistance should be calculated across the section at the mid-span of the column depth. Figure 15 shows a schematic of the second-order effects of the RC columns. Under the eccentric axial load N with initial eccentricity *e*0, the lateral displacement Δ*p* of the RC columns across the column depth at the midspan allowed the eccentricity of the axial load relative to the centre of mass of the section to reach Δ*p* + *e*0. Meanwhile, the moment of the RC columns increased from *M*1 = *Ne*0 to *Mmax* = *M*1 + *M*2 = *N* (*e*0 + Δ*p*), which is known as the second-order effect of eccentrically loaded columns [59,60]. During design, the second-order effects are accounted for using the moment augmentation factor [61].

**Figure 15.** Schematic diagram of the second-order effect.

According to [62], the relationship between the ultimate sectional curvature *ϕp* and Δ*p* of RC columns can be expressed as:

$$
\varphi\_p = \Delta\_p \left(\frac{\pi}{L}\right)^2 \tag{4}
$$

This equation is valid only if the first- and second-order deformations of a column can be expressed as sinusoidal shapes.

Therefore, the moment augmentation factor *η* can be expressed as:

$$\eta = \frac{e\_0 + \Delta\_p}{e\_0} = 1 + \frac{\varphi\_p L^2}{e\_0 \pi^2} \tag{5}$$

In the design code, an additional eccentricity *ea* is introduced because of uncertain load locations, uneven concrete quality, and construction variations, and a value of 20 mm is considered with an eccentricity of *ei* = *e*0 + *ea*.

#### *4.2. Bearing Capacity*

In general, the bearing capacity under eccentric compression is calculated using the equivalent rectangular stress diagram [61], assuming that the FITRC columns satisfy the plane section assumption, and the theoretical calculation can be performed according to GB 50010-2010 [61]. Furthermore, to simplify the calculation process, the tensile strength of the concrete was ignored. Figure 16 shows a simplified diagram for calculating the section of an RC column subjected to compression failure under a large eccentric load.

**Figure 16.** Simplified diagram for calculation of the section subjected to compression failure under large eccentric loading: (**a**) strain diagram of the specimens; (**b**) equivalent calculation diagram.

The compressive stress curve of the concrete in the compression zone was replaced with an equivalent rectangular diagram. According to the equilibrium conditions of the forces, the following formula was obtained.

$$N\_u = a\_1 f\_c b x + f\_{yc} A\_{sc} - f\_{yt} A\_{st} \tag{6}$$

Here, *α*1 is the equivalent rectangular stress block coefficient of concrete in the compression zone, defined as 1.0; *b* and *h* are the sectional dimensions of the RC columns; *fyt* and *fyc* are the yield strengths of the tensile and compressive longitudinal bars, respectively; *Ast* and *Asc* are the sectional areas of the tensile and compressive longitudinal bars, respectively. According to the resultant point of various forces on the tensile specimens in Figure 16b, the moment equilibrium conditions were determined and expressed as:

$$N\_{\rm u}e = a\_1 f\_c bx(h\_0 - 0.5x) + f\_{\rm yc} A\_{\rm sc} (h\_0 - a\_{\rm sc}) \tag{7}$$

$$e = \eta e\_i + 0.5h - a\_{st} \tag{8}$$

where *e* is the distance from the point of axial force to the resultant point of *Ast* of tensile longitudinal bars, *h*0 is the distance from the resultant point of *Ast* of tensile longitudinal bars to the edge of the compressive concrete, *x* is the depth of concrete in the compression zone, *ast* is the distance from the resultant point of *Ast* of tensile longitudinal bars to the edge of the tensile concrete, and *asc* is the distance from the resultant point of *Asc* of compressive longitudinal bars to the edge of the compressive concrete.

#### *4.3. Crack-Resistant Load*

According to the SL191-2008 standard in China [63], the crack resistance of RC columns under eccentric compression should be calculated as follows:

$$N\_{cr} \leq \frac{\gamma\_m a\_{ct} f\_t A\_0 \mathcal{W}\_0}{e\_i A\_0 - \mathcal{W}\_0} \tag{9}$$

where *γm* is the plastic section modulus, with the rectangular section set to 1.55; *αct* is the tensile stress limit coefficient of concrete, offset to 0.85; *A*0 is the area of the transformed section; and *W*0 is the elastic section modulus of the tensile edge of the transformed section, calculated as:

$$A\_0 = bh + \alpha\_E A\_{st} + \alpha\_E A\_{sc} \tag{10}$$

$$\mathcal{W}\_0 = I\_0 / (h - y\_0) \tag{11}$$

$$I\_0 = (0.0833 + 0.19\alpha\_E \rho) b h^3 \tag{12}$$

$$y\_0 = (0.5 + 0.425\alpha\_E \rho)h \tag{13}$$

where *I*0 is the moment of inertia of the column section, *y*0 is the distance from the axis of gravity of the transformed section to the compression edge, and *ρ* is the reinforcement ratio of the tensile longitudinal bars.

#### *4.4. Experimental Verification of Theoretical Predictions*

According to the aforementioned formulae, the moment augmentation factor and bearing capacity of the RC columns were calculated using the measured mechanical parameters of the rebar and concrete, respectively, and compared with the experimental results. As shown in Table 9, the theoretically calculated cracking load and ultimate bearing capacity were in good agreement with the measured results, with a maximum error of only 16%. In addition, there was a strength safety margin, which verifies that the calculation formulae are effective for FITRC columns under large eccentric loads.

**Table 9.** Bearing capacity of RC columns.

Table 10 lists the moments of the RC columns and the calculated moment augmentation factor. According to the structural design code [61], the moment augmentation factor *η* should be 1.0 when *L*/*h* ratios do not exceed 5.0. Furthermore, the moment augmentation factor of the FITRC columns is greater than that of the CRC columns in the control group. The measured moment augmentation factors of the six FITRC columns are close to each other, with a maximum value of 1.079, whereas the *η* value of the FITRC columns increases by a maximum of 1.5% compared to the CRC columns. This is because the *L*/*h* ratios of the RC columns in this study do not exceed 5.0 and the impact of the second-order effects is marginal.

In addition, Table 10 shows that the second-order moments of the FITRC columns are larger than those of the CRC columns. Furthermore, the average ratio of the second-order moments of the three FITRC45 columns to the CRC45 columns is 1.14, and that of the three FITRC35 columns to the CRC35 columns is 1.15. Therefore, if the second-order effect is to be considered in the structural design, the moment augmentation factor *η* of the FITRC columns should be 1.15 for safety reasons.

#### **5. Conclusions**

This study investigated the structural behaviour of FITRC columns under large eccentric loading. Six FITRC columns and two CRC column specimens were examined to investigate the effect of different raw materials, section dimensions, and eccentricities, on the mechanical behaviour of RC columns under large eccentric compression. The following conclusions were drawn from this study:

The ductility factor of FITRC columns is much lower than that of CRC columns, so the ductility of FITRC columns can theoretically be improved by stirrup reinforcement. However, this hypothesis requires further investigation.

**Author Contributions:** X.M.: Conceptualisation, methodology, validation, investigation, data curation, and writing of the original draft. J.S.: Conceptualisation, methodology, writing—review and editing, and supervision. F.Z.: Validation, data curation, resources, and supervision. J.Y.: Investigation and resources. M.Y.: Investigation and resources. Z.M.: Validation and visualisation. Y.B.: Investigation and resources. Y.L.: Investigation and resources. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Hebei Province Key Research and Development Project, China [grant number 19211502D], and the Hebei Province Graduate Innovation Funding Project, China [grant number CXZZBS2018109].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.


of states of the kind with which we are all familiar is more or less bound to continue in existence] because it is hard to see how the world could cease to be organized on a state-bound motif. The concept of freedom as non-domination that is associated with neo-republican theory provides a guiding ideal in the global, not just the domestic arena, and does so even on the assumption that there will continue to be many distinct states. It argues for a world in which states do not dominate members of their own people and, considered as a corporate body, no people is dominated by other agencies: not by other states and not, for example, by any international agency or multi-national corporation. (Pettit, 2016, pp. 47–48)

The influence of Pettit's work undoubtedly explains the reasons why this book examines the ideal of global citizenship education in order to distinguish various components (prescription, ideal, slogan, academic notion, educational ethics) and dig a Durkheimian path. To do so, the book tries to shed light, around this vast object, on all the themes addressed by Durkheim as an essential figure in sociology, philosophy and ETS.

This is why several investigations had to be conducted simultaneously: holism, collective entities, ideology of the modern, collective representations, organic solidarity, the state, individualism, cosmopolitanism. For to study citizenship and its education is necessarily to think of society itself, in all its components and in its aspect that is non-reducible to aggregated individual phenomena.

The stake is vast, but to have it fully in mind is already a first step, and this, with the help of the many works of neo-Durkheimian authors who, opening multiple and sometimes very different paths, have nourished the present reflections, from the nominalist micro-holism of liberal inspiration (Gilbert) to the holism displayed as being socialist. These elements help establish a crystallisation point, that of a cosmopolitan neo-republicanism at the height of sociological stakes. In this regard, one of the most crucial questions was: what status should the nation be granted?

Cosmopolitans in global citizenship education tend to respond that if citizenship is global, then nations are logically secondary elements. And why not? This position corresponds perfectly to the ideology of the modern, as Descombes, who remarked that

the normative individual, by definition, has no other social ties than those to which he has consented and from which he can always emancipate himself …. The true 'society of individuals' can only be, potentially, a global society. (Descombes, 2013b, p. 214)

Except that, according to Dumont and Descombes, one should not confuse the normative individual and the empirical individual, at the risk of suggesting that humanity is essentially composed of individuals, according to a sociological nominalism that is not really recognised but which, nevertheless, runs through the idea of global citizenship as soon as it is approached without the necessary reflective distance. Yet, the idea of global citizenship is available in the holist tradition in philosophy and sociology. In essence, a truly cosmopolitical citizenship education must make GCE a slogan, and it is the holistic tradition in philosophy and sociology that allow this. This perspective leads, as mentioned, to socialism, of which critical republicanism is a part.2

Thus, two pitfalls are avoided. The first one is that of permissible partialism, taking note of the fact that the preference for the nation is in tension with the moral equality of people in general: the identification with the nation, even patriotism, is then permissible strictly to the extent that it does not contradict cosmopolitan morality. The problem is that moral impartiality towards all is impossible to maintain. It would forbid us from preferring to help close family rather than the whole of humanity. More crucially, permissible partialism sees patriotism as a matter of individual preference. However, "to take patriotism as optional, or simply permitted, undermines the commitment to see the wrongs committed by one's own state as falling under one's own political responsibility" (Erez & Laborde, 2020, p. 194).

The second pitfall avoided is that of globally responsible nationalism, which echoes the conservative positions mentioned above (Miller, Schnapper). Of course, the nation is an ethical community and it is only through this path that individuals can be held responsible for the actions of their state (whether single or pluri-national). If it commits injustices in the global framework, its subjects must feel connected to it in order to counter its actions. Globally responsible nationalism would therefore seem an interesting middle way between abstract cosmopolitanism and narrow nationalism, but the problem is that strong identification with the nation is a mechanism that reinforces the denial of injustices. One thinker will produce a narrative saying that the French Empire is not really responsible for the current problems of formerly colonised countries, while another will claim that the dominant national group is in fact the real victim.

The neo-republican approach is more nuanced. Of course, it is the belonging to a political society that awakens the commitment to defend individual freedoms. In this sense, neo-republicans are patriots. They echo Durkheim's intuition that it is modern societies as such, marked by the integrated and regulated division of social work, that celebrate the cult of the human person. This cult is therefore a collective ideology, even if its object is the individual. Nations, in turn, are judged by individuals according to this collective ideology; 210 Conclusion

that is why criticism of the nation does not imply the rejection of patriotism but is a constitutive part of it. Neo-republicanism nevertheless pays attention to groups' claims that they are dominated because it defends, more precisely, the idea of an enlightened self-interest. This is crucial: the best way to defend one's own freedom is to support public institutions that defend the freedom of all.

This neo-republican path, which poses at the same time the necessity of what is collective, or even societal (patriotism), and the critical skills resulting from our modern ideology (the cult of the individual that Durkheim spoke of) is thus also a neo-holist path. This is the missing piece to dispel the confusion that haunts part of the debates on GCE, between nominalism and conservatism. Acknowledging the urgency of clarifying the complex relationships between national and cosmopolitan ideals, social and political philosophy offers social sciences a lexicon referring to collective entities that frame republican principles, while erecting safeguards against any instrumentalisation of these entities in the service of an intellectual and political agenda that undermines individual freedoms. Thus, the cosmopolitan patriotism that Durkheim called for is updated. Its relevance has been confirmed by the Globalsense research program, and could lead to other avenues of work in education and training sciences.

#### **Notes**

Dubuisson, D. (2019). *The invention of religions.* Equinox Publishing.

Fabiani, J.-L. (2016). *Pierre Bourdieu. Un structuralisme héroïque*. Seuil.

Mauss, M. (1968), *Essais de sociologie*. Seuil.

Rosanvallon, P. (2006). *Democracy past and future*. Columbia University Press.

## **Index**

128, 129, 131, 134, 137, 139–141, 143, 146, 158–161, 164–173, 175–177, 180–182, 184, 186, 188–194, 200–202, 204, 205, 207, 209, 210

## **Global Citizenship Education Modern Individualism under the Test of Cosmopolitanism**

### **Sébastien Urbanski and Lucy Bell**

Global citizenship education is an essential topic in an increasingly interconnected world. Indeed the need for inclusive and globally conscious education, embedded in cosmopolitanism, is recognised as a way to prepare individuals to navigate diverse cultures, address global challenges, and actively participate in a globalised world.

Being both scientifijic and political, these challenges require an interdisciplinary exploration of citizenship education, merging sociology, philosophy, as well as education and training sciences. To do this, *Global Citizenship Education: Modern Individualism under the Test of Cosmopolitanism* offfers a framework that integrates Durkheim's holistic approach with critical republicanism.

The book is also rooted in the analysis of data collected through GlobalSense, a research project that focuses on preparing teachers to navigate the complexities of GCE within an international context. By presenting both a theoretical reflection and an analysis of an international training program within universities, this book can be of interest to academics, teacher trainers and (future) teachers themselves.

**Sébastien Urbanski**, Ph.D. (2012), Aix-Marseille University, is an associate professor at Nantes University. He has recently published a book on secularism and discrimination in school contexts, *Laïcité, discriminations, racisme: Les professionnels de l'éducation à l'épreuve* (with Françoise Lantheaume, Presses Universitaires de Lyon, 2023).

**Lucy Bell**, Ph.D. (2019), Nantes University, is a postdoctoral researcher at Nantes University. She is the scientifijic coordinator of the GlobalSense research, funded by the European Commission and the Pays de la Loire Region.

Cover illustration: iStock.com/Yakobchuk

CAIEFR 1 ISSN 2949-8805

> Spine <14 mm>

**Sébastien Urbanski and Lucy Bell**

**Global Citizenship Education**

**Global Citizenship**

**Modern Individualism under**

**the Test of Cosmopolitanism**

**Sébastien Urbanski and Lucy Bell**

**Education**


**1. Introduction**

Graphs are becoming ubiquitous across a large spectrum of real-world applications in the form of social networks, citation networks, telecommunication networks, biological networks, etc. [1]. In addition, numerous applications involving multimedia, such as video surveillance, video streaming, healthcare systems, and intelligent indoor security systems depend on using graphs as research objects [2–4]. For a considerable number of real-world graph node classification tasks, the training data follow a *long-tail* distribution, and node classes are *imbalanced*. In other words, each of a few "majority" classes has a large number of samples, while most classes only contain a handful of instances. Taking the NCI chemical compound graph as an example, only approximately 5% of molecules are labeled as active in the anticancer bioassay test [5]. Graph node classification tasks are often further complicated by the fact that graph nodes can be associated with multiple labels in many real-world network data. Many social media sites, such as BlogCatalog, Flickr, and YouTube, allow users to use a diverse set of labels representing their various interests. A person can join several interest groups on Flickr, such as *Landscape* and *Travel*, and different

**Citation:** Duan, Y.; Liu, X.; Jatowt, A.; Yu, H.-t.; Lynden, S.; Kim, K.-S.; Matono, A. SORAG: Synthetic Data Over-Sampling Strategy on Multi-Label Graphs. *Remote Sens.* **2022**, *14*, 4479. https://doi.org/ 10.3390/rs14184479

Academic Editors: Jungho Im and Gwanggil Jeon

Received: 26 June 2022 Accepted: 26 August 2022 Published: 8 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

video genres on YouTube, such as *Cooking* and *Wrestling*. Furthermore, many networks are characterized by imbalanced label distribution and multi-label nodes at the same time, as shown in Figure 1.

**Figure 1.** Label distribution of three real-world multi-label network datasets: (**a**) BlogCatalog3 [6], (**b**) Flickr [6], and (**c**) YouTube [7]. The horizontal axis represents the label id, and the vertical axis represents the number of nodes containing each label. It can be clearly observed that these label distributions are all highly imbalanced, where a few classes contain many more nodes than the rest of the classes.

To date, a large body of work has focused on graph representation learning (GRL) with balanced node classes and simplex labels [8–12]. However, these models do not perform well when graphs exhibit the aforementioned characteristics of being imbalanced and multi-label for the following reasons. (1) *The problem caused by the imbalanced setting*: The imbalanced data make the classifier overfit the majority class, and the features of the minority class cannot be sufficiently learned [13]. Furthermore, the above problem is aggravated by the presence of the *topological interplay* effect [5] between graph nodes, making the feature propagation dominated by the majority classes. (2) *The problem caused by the multi-label setting*: Multi-label graph architectures typically encode very complex interactions between nodes with shared labels [5], which is challenging to capture. Therefore, it is essential to develop a specific graph learning method for class imbalanced multi-label graph data. However, research in this direction is still in its infancy. Thus, in this study, we propose *imbalanced multi-label GRL* to address this challenge while also contributing to graph learning theory.

For imbalanced data, *minority over-sampling* is an effective measure to improve the classification accuracy [14–16]. This strategy has recently been confirmed to be effective for graph data as well [17]. Traditional over-sampling techniques consist of a two-step process: (1) the selection of some minority instances as "seed examples"; (2) the generation of synthetic data with features and labels similar to the seed examples, which are then added to the training data. For example, the most popular over-sampling technique— the synthetic minority over-sampling technique (SMOTE) [14]—addresses the problem of minority generation by performing interpolation between randomly selected minority instances and their nearest neighbors. Cost-sensitive learning is another type of effective approach for alleviating the problem of imbalanced data applied to a classification [16], where the basic assumption is that the cost resulting from different types of misclassification varies significantly (e.g., the cost of treating an intruder as a non-intruder is much greater than treating a non-intruder as an intruder). The principle of applying cost-sensitive learning methods to imbalanced learning problems is to assign a larger penalty cost to misclassified minority class samples [18]. Existing cost-sensitive classification algorithms can generally be grouped into three categories [19]: algorithms that (1) pre-process the training data, (2) post-process the output, and (3) apply direct cost-sensitive learning methods. Data pre-processing aims to make the classification results on the new training set equivalent to cost-sensitive classification decisions on the original training set, typically

along the lines of sampling [18] and weighting [16]. Post-processing the output makes the classifier biased toward minority classes by adjusting the classifier decision threshold, as represented by MetaCost [20] and ETA [21]. Direct cost-sensitive learning methods embed the cost information into the objective function of the learning algorithm to obtain the minimal expected cost, such as cost-sensitive decision trees [22] and cost-sensitive SVM [23].

However, mainstream over-sampling techniques have significant shortcomings when applied to graph data, as the selection of seed examples prioritizes global minority nodes while ignoring local minority nodes, and each synthetic instance is always assigned a label based on some specific strategy, which may be incorrect. This is because, in contrast to non-graph data, the relationships between graph nodes are explicitly expressed by the edges connecting them, meaning that the representation learning of a node can be heavily dependent on its neighboring *unlabeled* nodes through the feature propagation mechanism inherent to graphs.

Motivated by the above observations, we propose and test the following hypothesis. In addition to synthetic minority samples, synthetic *unlabeled* samples can also facilitate the debiasing of graph neural networks (GNNs) on an imbalanced training set. In particular, for nearby global minority samples that are a local majority, we can "safely" produce virtual samples of the same class and add them into the training sets to balance the class distribution. Global minority samples that are also a local minority are more likely to be local outliers, and thus, they are risky for selection as seed examples for further oversampling. For nearby global minority samples whose neighbors are class-balanced, it is difficult to determine the labels of virtual samples. Thus, the production of *unlabeled* virtual nodes should be encouraged, which can help minorities by "blocking" the overaggregation of the majority features delivered through edges. This idea is illustrated in Figure 2. **We argue that the key to over-sampling on an imbalanced multi-label graph is to flexibly combine the synthesis of both labeled and unlabeled instances enriched by label correlations.**

**Figure 2.** A comparison between our method and the current state-of-the-art graph over-sampling method **GraphSMOTE** [17]. In the current method, the idea is to generate new minority instances near randomly selected minority nodes and create virtual edges (dotted lines in the figure) between those synthetic nodes and real nodes. Instead, we synthesize minority instances in safe areas (i.e., A1), generate *unlabeled* instances in locally balanced areas (i.e., A2), and do not conduct data oversampling near minority nodes that are outliers (i.e., A3). For the simplicity of illustration, only a single-label scenario is shown.

We extend the existing over-sampling algorithms to a novel framework for the imbalanced multi-label graph node classification task based on the above considerations. We extend the classic global minority-based seed example selection to the local minority perspective (see Section 3.2). Distinct from interpolation, which is commonly used in mainstream over-sampling techniques [24], we use a generative adversarial network (GAN) [25] to generate new instances. As a representative deep generative model, a GAN can capture label correlation information by estimating the probability distribution of seed examples [26]. We propose an ensemble architecture of a GAN and conditional GAN (CGAN) [27] for the flexible generation of both unlabeled and labeled synthetics (see Section 3.3). To make use of the graph topology information, we propose a method to obtain new edges between the generated samples and existing data with an edge predictor (see Section 3.4). The augmented graph is finally sent to a graph convolutional network (GCN) [12] for representation learning, together with the learned label correlations (see Section 3.5). We name our proposed framework **SORAG**, abbreviated from **S**ynthetic data **O**versampling St**RA**tegy on **G**raph. In summary, our contribution is three-fold:


**3.5 Rationale**

The mission of Prepare2Nspire is to prepare marginalized students to succeed on grade-level and high-stakes mathematics exams and to inspire them to continue their *Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

study of mathematics. This mission is accomplished by developing mathematics confidence, content knowledge, connections, communication skills, and community through its cascading tutoring and mentoring model. P2N also works to create a STEM pipeline for urban underrepresented students to post-secondary education and opportunities.

Prepare2Nspire uses the terms *mentutor* and *mentutee* to signal the combination of the mentoring and tutoring roles. Recognizing that mentoring is a function of building relationships and that tutoring entails the process of assisting in problem-solving and working through mathematics content, the program merged the roles of tutor and mentor and the roles of tutee and mentee. Tutoring without mentoring, however, removes the important leverage of building relationships. Mentoring mirrors tutoring, in the beginning, because it passes the expertise of the tutor onto the tutee. Furthermore, mentoring cultivates both study skills and positive behavior, creates resilience and self-reliance, provides context for the exchange of information and knowledge (which, in the case of P2N, bridges the gap between mathematical theory and practice), and cultivates leadership competencies in both mentor and mentees. For mentees, it also increases the capacity for service to others in the future just as their mentors are currently serving them.

#### *3.5.1 Culturally responsive mentoring*

Prepare2Nspire works to incorporate culturally responsive mentoring into the daily lives of students as a way to empower and strengthen the relationship between tutors (mentutor) and tutees (mentutee). Particularly troubling is the myth that culture does not matter in the teaching and learning of mathematics [31], especially when many states continue to have large academic and opportunity gaps between students of color and white students [32]. Paying attention to "issues of race and culture in the way we teach mathematics has incredible power to disrupt the troubling opportunity gap" [31]. Using what students already bring to class acknowledges that such issues exist and addresses them by providing a strategy to mentor African American students from urban schools.

#### *3.5.2 The shaping of mathematical identity*

Small learning communities form the foundation of Prepare2Nspire. These table groups are named for underrepresented STEM scholars like Mae Jemison, an African American astronaut and scientist, and Katherine Johnson, an African American mathematician known for her computation work for NASA. The naming of the communities using underrepresented scholars is intentional. To help students see themselves as people who can excel in science and mathematics, they should see and know previous scholars who have succeeded in STEM fields this way.

While mathematical identity is a social construction [33, 34], participating in mathematics classes has a profound effect on the development of that identity. Racialized narratives [35, 36] are among the factors that influence it. More often than not, schools have perpetuated deficit narratives by placing students from underrepresented populations into remedial courses that make it nearly impossible to participate in advanced study. Students, in turn, begin to identify as being incapable and unable to perform academically, believing the implicit message that they aren't capable of doing and being better. Such messages are internalized, creating these deficit narratives and low self-worth even when students have the skill set or prior experiences to demonstrate they can be successful. These "[deficit] identities intersect with already

existing stories about other kinds of social identities" [35, 37, 38]. P2N pushes back against this notion by transforming student mathematical identity.

Cultivating positive peer relationships [39, 40] through the use of learning communities is an approach to influencing the formation of mathematical identity. Lieberman [41] asserts that learning among low achieving students increases when they are placed in empowering roles. Situated learning suggested by Lave and Wenger [42] is a model that places learning within social relationships. Thus, community participation is imperative when cultivating mathematical identity, and positively affects the mathematical identity of the tutee [43]. The P2N near-peer model is an example of mathematics support situated in a social setting. Positive encouragement from peers is an effective and motivating strategy. As one P2N participant states, "I really like how when we come here we build community, and at the same time we're also learning [math]. And we meet people who are from different backgrounds" (personal interview, December 2017).

#### **3.6 Structure of Prepare2Nspire**

#### *3.6.1 "Prepare": reverse the current trend*

In almost every accountability measure in Minnesota (Minnesota Comprehensive Assessments MCA-II and Minnesota Comprehensive Assessments MCA-lll) and nationally (National Assessment of Educational Progress—NAEP), certain groups of students consistently lag behind their White peers [44]. African American, Native American, and Hispanic students, as well as students from low socioeconomic backgrounds, perform behind their peers in academic preparation, high school graduation, and college attendance and completion. In addition, these students are often enrolled in fewer STEM courses in high school and college. The STEM job market is growing, with specialized fields that will not include students from the aforementioned groups if they do not have a background in mathematics and other STEM courses. Studies by NAEP and Jett [45] reveal that the "overwhelming number of low-achieving students in algebra are black and Hispanic and attend big urban, highpoverty schools where they are more likely to fall through the cracks" [46] and that African American and Hispanic students are disproportionately underrepresented in advanced mathematics courses. Jett [45] and others [35] assert that the highest predictor of college readiness and completion is the taking of higher-level mathematics courses during high school. Prepare2Nspire is located in an urban community where mathematics failure is common. If the trend of disproportionate underrepresentation of marginalized students is to be reversed, foundational math failure must be made a high priority in urban educational settings where the numbers are most staggering. Succinctly, prepared students can reverse this trend.

#### *3.6.2 "2": the participants*

Prepare2Nspire was developed to support two cohorts of students: eighth graders and eleventh graders. Students in these cohorts are referred to as mentutees, since they are recipients of both mentoring and tutoring [47]. The design of the program is strategic, with middle school and high school students seeing college undergraduates pursuing higher education especially in mathematics or other STEM fields. These undergraduates, called mentutors, are also active participants in the program as they develop different skill sets like leadership and teaching (**Figures 4** and **5**).

*Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

**Figure 4.** *Prepare2Nspire.*

**Figure 5.** *Prepare2Nspire at UROC Spring 2017.*

The original design of the program matched one undergraduate with three eleventh graders and six eight-grade participants. As the program has developed and evolved over the last 10 years, outside circumstances may have altered the exact number of students from each cohort within a community, but the nearpeer model has only become stronger, regardless of the changing number of participants. The undergraduate remains the foundation of each community while supporting both eighth and eleventh graders and as eleventh graders support eighth graders.

#### *3.6.3 "Nspire": the role of technology*

Graphing calculators have transformed how students think mathematically in a classroom and on assessments. Prepare2Nspire is influenced by the graphing calculator with the same name. Each participant receives a graphing calculator and is taught how to use it effectively. Since mathematical thinking is influenced by technology, and technology can be used on standardized assessments, participants are taught how to move between the multiple representations which are incorporated into the teaching and learning of mathematics in general, and in algebra specifically. Given that technology is an integral part of STEM occupations, this tool can be an advantage when preparing for this pathway IF students know how to use it.

When students attend each week at their table communities, they work, eat and talk with the same people. All of these activities are an intentional part of building relationships which, in turn, enable authentic conversations about algebra. The sense of community transforms how participants think about mathematics.

#### **3.7 The immersive maker space program**

Building on prior literature and experiences from the supporting programs Building Bridges to Design and Prepare2Nspire, we used an inclusive lens to create and deliver immersive maker space programs for underrepresented students in grades 4–12 in summer 2020, 2021 and 2022. The project-based making exercises focused on the intersection of design and mathematics, drawing on historic design precedents from ethnic minority communities including African, African American, Hmong, and Vietnamese communities, where underrepresented student participants in this program came from. Using design precedents from diverse communities helped our team create a "culturally-responsive approach" that is inclusive and relevant to student participants. We exposed K-12 students to design, engaged them in hands-on experiences, and created opportunities for them to collaborate with underrepresented mentors. We also used design precedents from underrepresented designers to illustrate contributions to the built environment.

Through extensive literature review on maker spaces and underrepresented students, focus group meetings, ideation and mockup sessions with University researchers, and collaboration with community and school partners, we co-designed and created a curriculum for an immersive maker space in a city with a poor record of inclusive excellence. Our goal was to offer hands-on experiences in design, 3D modeling, online experimentation, making, and digital fabrication to promote STEM/ STEAM learning. The immersive maker space was implemented online in 2020 and 2021 due to the COVID-19 pandemic and in-person in 2022 with grade 4–12 students at the Robert J. Jones Urban Research and Outreach-Engagement Center (UROC) in

*Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

North Minneapolis, a school zone with an exceptionally high educational achievement gap. Throughout the program, students' learning was measured through survey responses and informal interviews.

## **3.8 Program curriculum and activities**

The program curriculum included College of Design students from diverse backgrounds sharing their career interests and projects via videos. The design and STEAM videos included introductions to fractals, geometric principles, the golden rectangle, and the Fibonacci sequence. Fractals in design and architecture were presented through a cross-cultural lens, showcasing fractal geometry in the work of underrepresented designers like David Adjaye, including his design of the African

#### **Figure 6.**

*Summer 2020 student sketching, paper folding, and LEGO and building blocks projects.*

#### **Figure 7.** *Summer 2020 3D modeling in TinkerCAD and paper folding projects.*

#### *Pedagogy, Learning, and Creativity*

American Museum in Washington DC. For the introduction to mathematics, graphing calculators, and coding, students engaged in hands-on learning about how to code and automate devices. Hands-on paper folding exercises were integrated so students could learn about various math principles while folding structures and origami forms.

For sketching, 3D modeling, and 3D printing, students were introduced to 3D modeling in TinkerCAD using Cartesian coordinate systems, points, lines, and basic geometric shapes such as cubes, planes, and spheres. They learned rapid prototyping through 3D printing of the TinkerCAD models they created. For example, they created massing models of the African American Museum in DC in TnkerCAD. They learned about the massing and composition of the structure, which are extruded trapezoidal geometric shapes stacked vertically to form the building facade. This exercise provided students with the opportunity to learn about the intersection and relationship between mathematics and design principles.

**Figure 8.** *Summer 2022 3D modeling in TinkerCAD and 3D printing.*

**Figure 9.** *Summer 2022 sketching and LEGO exploration.*

*Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

For LEGO and building block modeling, students experienced and learned about design and mathematics by building models of global buildings, including the Khufu Pyramid, the Sydney Opera House, the Tower Bridge, the Pisa leaning tower, and skylines in Japan and Dubai Skyline (**Figures 6–9**).

## **4. Discussion and findings**

To support our goal of implementing the maker space program and engaging underrepresented students to promote future success in STEAM, we measured students' learning throughout the program using surveys and informal interviews. Students completed pre-surveys prior to participating in the program and post-surveys after the program. To help the authors learn about the effect of the program on student learning, attitudes, and engagement in the program, sample survey questions were targeted towards asking participants about their experience before and after the camp. For example, participants were asked about the design and mathematics careers they know, the STEAM skills they were learning, and their best experience in the program (**Figure 10**).

The 2020 camp had to be delivered virtually due to the COVID-19 pandemic lockdown. From the 2020 cohort of 64 campers (34 grade K–6 and 30 grade 7–12 students), the majority of the campers—81.5%—considered the camp informative and interesting on a Likert scale of 1–5 points. No camper rated the program as not informative or interesting. When asked what they learned in the virtual summer camp, students reported learning about design. For example, one participant quotes "you can start designing and drawing and someday that dream of yours might come true." Another noted "I learned about the different types of designs people do on a daily basis which made me into design." Another participant noted "I learned about possible career paths in design. I also learned about certain buildings." When asked about the best experience in the virtual summer camp, participants reported enjoying the virtual design firm visits, origami, and LEGO building. One participant reported "The best part was when people that do different types of designs and when they showed us what they do every day on the job and explained how things work at their

**Figure 10.** *Word cloud illustrating best experience in Summer 2021.*

offices." Another reported "Building the Origami helps me think better." Another reported "I really liked how challenging the LEGO building was."

The 2021 camp was also held virtually due to the COVID-19 pandemic. From the 2021 cohort of 30 campers (18 grade K–6 and 12 grade 7–12 students), the majority of the campers—72.2%—considered the camp informative and interesting on a Likert scale of 1 to 5 points. No camper rated the program as not informative or interesting. When asked what they learned in the virtual summer camp, students reported learning about mathematics and design. For example, one participant noted "I learnt that math and art have a good way in incorporating itself in designing projects." Another noted "Fractals is a concept in math that deals with space and dimensions." On modeling, another participant noted "I learnt that 3D printing is sometimes used to small-scale a building like a model." When asked about the best experience in the virtual summer camp, participants enjoyed the origami, sketching, LEGO building, and modeling exercises. For example, participants reported their best experiences as follows: "seeing the different people and what branches of engineering they worked in," "trying to build the Japanese skyline," "drawing the fractals and building the origami," and "I liked the crafts and the origami, and it was interesting."

The summer 2022 camp was delivered in-person. For this summer experience, the Arts were included in the STEM focus, so the focus was Science, Technology, Engineering, Arts and Mathematics (STEAM). The program followed Minnesota COVID-19 protocols to ensure a safe environment, and all participants wore masks during indoor activities. Since the camp was delivered in-person, more detailed preand post-surveys and informal observations were conducted. From the 58 registered campers, 23 attended the full 2 weeks of the summer camp and participated in the pre-and post-survey (7 grade K–6 and 16 grade 7–12 students). When asked in the post-camp survey how the experience would contribute to their future plans, participants reported wanting to become engineers, architects, entrepreneurs, fashion designers, computer programmers, and animators. Notably, a participant reported that the program taught "me how to create and provide helpful things to the economy." Additionally students reported that the experiences helped them with mental mathematics, technology, and logical skills. These findings, demonstrating stronger STEM/STEAM self-efficacy (**Figures 11** and **12**), support the program directors' intentions to use creative project-based learning exercises to promote future success. **Figures 11** and **12** summarizes the average Likert score of summer camp participants attitudes towards STEAM pre-and post-camp. For example, as illustrated in the diagrams the average Likert score for the following questions increased from the pre to post: *I think I can do well in STEAM (3.82 to 4.45), I think STEAM will help me even when I am not in school (3.73 to 4.00), I am interested in thing I learn in STEAM (3.73 to 4.05) and I enjoy doing STEAM projects (3.70 to 4.35).*

As seen in comparison of **Figures 11** and **12**, most participants reported that the program helped them understand STEAM as much as they can. The average Likert score of the other five questions shows an increase after the camp, indicating an increase in students' interest in the STEAM class. All students thought STEAM is important and enjoyed doing STEAM projects. Through the camp, they mastered many new skills, and believed STEAM would help them even when they are not in school. The summer design camp helped them improve their interest in STEAM. However, it looks like students were not looking forward to their STEAM classes in school. A paired-sample t-test was conducted to the pre-and post-test survey questions to determine if there was significant difference at the end of the camp experience. As

*Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

#### **Figure 11.**

*Pre-Survey of 2022 Summer Camp participants attitudes towards STEAM.*

#### **Figure 12.**

*Post-Survey of 2022 Summer Camp participants attitudes towards STEAM.*

#### **Table 1.**

*Student Interest in STEAM pre- and post- summer camp. Mean, Standard Deviations for Pre- and Post-test data and t Test results (n = 23).*

shown in **Table 1** there was significant difference in pre-and post for the question *I enjoy doing STEAM projects*. Our limitations are the small sample size in the in-person summer 2022 camp due to the COVID-19 pandemic. A larger sample size will be needed to show any statistically significant difference before and after camp.

In terms of the program directors' goal to close the learning gaps due to the COVID-19 pandemic, the hands-on experiences in 3D modeling in TinkerCAD and in coding and 3D printing provided the opportunity to tackle digital inequities and the lack of access to current technologies faced by underrepresented students. Overwhelmingly, student participants mentioned 3D printing, modeling, and coding when asked about their best experience post-camp.

In terms of the goal to use an inclusive lens of STEAM learning, the projectbased curriculum used a "culturally-responsive approach" incorporating design examples from underrepresented and minority designers to teach about fractals, geometry, design, and architecture. The visual and design examples came from patterns, visuals, architecture, and ornamentation from African, African American, Hmong, Vietnamese and global cultural examples, helping student participants see role models and a reflection of themselves in the project materials.

## **5. Conclusion**

Findings from the program across the years indicate a high level of engagement and positive learning experiences among program participants. By learning about design and mathematics, participants gained broader perspectives and discovered new interests and career paths. Additionally, our findings highlight how understanding the connections between different disciplines helps foster creativity and

#### *Bridging the Digital Divide in Design and Mathematics through an Immersive Maker Program… DOI: http://dx.doi.org/10.5772/intechopen.111787*

innovation. Our findings are consistent with those of authors such as Gauntlett [29], that found LEGO experiences offer students creative thinking opportunities [29], and Marsh et al. [22], that highlighted the importance of exposing underrepresented students to maker spaces [22]. Similar to previous authors, we found that anecdotal evidence from this program showed underrepresented students learned about their cultural roots from the program experiences [23].

As the UMN student body is not diverse, programs like this can provide pathways to future careers and bridge educational disparities. The program instills in participants the motivation to continue exploring these disciplines, and helps them develop relevant knowledge and skills for the future. Universities and educational institutions play a critical role in promoting diversity, equity, and inclusion in education and future careers. Programs aimed at increasing the representation of underrepresented students, such as those from low-income or minority backgrounds, can reduce educational disparities and provide these students with the skills and resources they need to succeed in their chosen fields. By creating inclusive and supportive environments, universities can help foster a sense of belonging and empowerment for underrepresented students and help them reach their full potential.

The COVID-19 pandemic caused widespread learning losses, particularly among students from underrepresented communities who may have faced additional challenges such as limited access to technology and disrupted home environments. To bridge these learning losses, this program introduced initiatives such as:

It is important to address the learning losses caused by the pandemic and to ensure that all students have access to the resources they need to succeed in their education. By working together, educators, policymakers, and communities can help mitigate the impact of the pandemic and ensure that students have the opportunities they need to reach their full potential.

One of the biggest challenges in carrying out the above initiatives is funding. Scaling up these initiatives to reach more underrepresented students and have a greater impact requires significant financial resources. It is crucial to find innovative and sustainable funding models that can support the scaling up and larger impact of these initiatives. This could include partnerships among government, business, and non-profit organizations, leveraging existing infrastructure and resources, and exploring alternative funding sources such as crowd-funding and social impact bonds. Ultimately, the key to overcoming funding challenges is a combination of creativity, persistence, and collaboration. By bringing together stakeholders from different sectors and working together towards a common goal, it is possible to secure the funding needed to provide access to digital experiences, technologies, and digital fabrication for underrepresented students.