6. Results and Discussion
Using the statistic method described above, we analyzed whether the educators’ population had an integrated practice based on the 15 features of population proportions described in the last section. The results are shown in
Table 5.
Due to the above empirical rule and as shown in the shaded columns of
Table 5, for questions NCKP-P1, NCKP-P4, PKNCP-P1, PKNCP-P2, CKNCP-P1, CKNCP-P2, and RP-P2, it was not possible to assume normality. In these cases, the proportions tended to take extreme values. In this situation, it is natural to discard these categories because this demarcates a trend as to whether these practices were “observed” (∼1) or “not observed” (∼0).
For the rest of the categories, it was possible to assume normality, and this allows us to reinforce the point estimate shown in
Table 5. Therefore, we calculated the eight confidence intervals using, as is usual, 95% confidence. The results of this are shown in
Table 6, where we detail the point estimates
(where
i = 1,…,15 are the features of each category, labeled from up to down in
Table 5) and the lower limits
and upper
for each of them. Once the confidence intervals were calculated, the margins for the confidence intervals were within reasonable values.
The NCKP-1 and NCKP-4 features were not integrated into the NCKP category of teachers’ teaching practice. This could imply that pre-service educators do not have enough solid formal structure of the school knowledge that they are teaching. In particular, the reported practice does not reflect discussions on building the concept of numbers in preschool mathematics. Similarly, the communication of mathematical results through multiple representations, both concrete and abstract, is not an activity that educators demand of children. However, most lesson plans reflect good work with children on the basic principles of counting and number reasoning, considering the child’s logical thinking. These last features correspond to NCKP-2, NCKP-3 (weakly integrated), and NCKP-5 of the NCKP category.
Once the organization of (scholar) math objects is clear to the subject who teaches, he/she must build multiple representation schemes that help design communication channels with the subject who learns. At the preschool level, a vital vehicle to achieve a level of understanding with the child is the game’s didactic strategy, which must be regulated into the design class structures to keep accurate records of the child’s learning. A frequently used alternative applied by teachers is qualitative analysis through observation guides in the classroom on activities with specific intentions about what the educator wants to observe and how he wants to relate it in their general intervention plan.
This corresponds to PKNCP-3 (weakly integrated) and PKNCP-4 features. The multiple representations of the mathematical objects are integrated into the child’s games, and the educator will present them. Here, the tangible materials that the educator designs are part of the communication channel that the child establishes with the object of knowledge and not only with the educator or with other children. The teaching materials’ function should be aligned with the educator’s purposes in his/her lesson plan. The connectivity and relationship between these elements must be transparent within the lesson plan and, particularly, in the didactic sequence described by the educator as indicated by the PKNCP-5 (weakly integrated) feature.
In the medium or long term, the educator’s records about their teaching intervention place them in a suitable position to describe and understand how their students learn. In other words, he/she empirically has constructs and theories about learning induced by their teaching practice that are still in an unconsolidated development of PKNCP-1 and PKNCP-2 features.
In a way, both the NCKP category and PKNCP maintain certain independence until analyzed together with the CKNCP category. It is possible to assume that the educators’ practice is dominated by intuition and the experience accumulated in their (short) teaching practice but not by considering theoretical positions that help them adequately develop their intervention. Likewise, it is observed that an intervention is performed without a registered (in the teaching practice reports) work methodology that places them in a position to improve the practice itself.
The congruence of educational practice is strictly determined by the curriculum that governs the teaching–learning process when we speak of socially schooled systems. Through the curriculum that pre-service educators use to conduct their practice, the instrumentation of the curriculum requires compliance with specific control standards as set out in the profile of graduation from preschool education. Therefore, it is essential to consider that the hidden (or lived) curriculum plays a fundamental role in teacher development when innovating their educational practice.
However, for this to happen, the educator must bring together the NCKP and PKNCP categories and align them with CKNCP, but how is it possible to perform such an undertaking if there is no evident integration of the NCKP, PKNCP, and CKNCP categories? This inevitably leads to a practice detached from the philosophical foundations and principles of a given curricular proposal as reflected by the absence of CKNCP-P1 and CKNCP-P2 features in their teaching practice.
Despite this, the organization of learning within the class plans of the analyzed educators shows that the structural aspects of the study plan with which the practice is accompanied are known, highlighting that, in general, they have CKNCP-P3 features. This indicates correct use of the curricular organizers and moderately acceptable use of the expected learning targets.
The conscious integration and alignment of the NCKP, PKNCP, and CKNCP categories cannot occur without the RP category. To describe how pre-service educators understand and interpret RP, we must address three fundamental questions: (i) How do they understand and perceive the process of reflection during educational practice? (ii) Do they describe how they learn to reflect on their practice? (iii) Under what context does the reflection occur?
Reflection proceeds the intelligent, active, persistent, and careful action of any supposed form of knowledge considering the motives that support it and the consequences to which it leads. If we refer to question (i), we immediately notice that the construct of educational reflection depends, to a great extent, on the discipline that is taught. This is because the NCKP category’s domain determines the educational intervention’s perception based on what is observed by the educator about the children’s mathematical reasoning. This, in turn, allows him/her to modify the practice of it.
The iterative cycles that can occur within a sequence made up of numerous lesson plans would eventually make the educator perceive a reflection process if he/she notices and records changes in their educational practice. Thus, he/she would gain awareness about the educational process and would be able to perceive educational reflection. However, this implies having the ability to look back in practice critically and imaginatively, performing task analysis, and looking ahead through purposeful planning. This intelligent action that defines the RP is a social object that results from interpreting the teaching–learning dynamic processes.
In other words, a reflection occurs in the action of attempting to understand the relationship of the teaching object, the learning subject, and their role as a teaching subject. These features would help pre-service educators to describe how they reflect, based on the integration of the NCKP and PKNCP. The teaching experiences, situated in socially specific contexts, incorporate the CKNCP dimension to synthesize the educational act’s understanding, thus, originating a personal vision of the entire teaching–learning process.
The RP allows understanding and controlling the content and processes of their work in the classroom, helping the educator to control what happens in the teaching process and to make decisions that define the educational path in which he/she is immersed. The reflection occurs in specific space–time coordinates that give rise to a reflection linked to the concrete action that the educator performs in their working environment. In this way, question (iii) brings us closer to thinking that the RP is a deliberate and systematic investigation of practice that impacts the detection and development of students and of itself.
Hence, the RP is of great interest and importance. It is the amalgamation between teaching and learning or, in other words, of the NCKP, PKNCP, and CKNCP categories. In the context of teaching the concept of numbers in preschool, the RP added to the NCKP makes it possible for the educator to adequately reason regarding their lesson plans and to make them intentional and reflective, linking them to the preschool graduation profile and the entry profile of primary education. Let us not lose sight of the fact that the dimensions mentioned at the beginning of this document are general and that, here, the only example under discussion is the construction of the concept of numbers in preschool at its various levels of development.
The content analysis results that were shown are based on the categories CK, PCK, CCK, and RP defined a priori in this study. These categories applied to the context of teaching the concept of numbers in preschool allowed us to understand the integration of these elements in the teaching practice of the pre-service educators that composed our study sample. However, within the content analysis technique, the analysis guide that emanates from these categories is not governed by any rule that allows deeper investigation. In this study, the said guide mapped each lesson plan analyzed to a row vector of 15 entries with values 1 or 0 depending on whether the previously established traits for each category were observed or not in the units of analysis.
In this way, a certain number of those 15 entries corresponded to a specific category that, in short, accounted for the level of integration of that category of knowledge in the educator. From now on, what we propose in this additional analysis of our results is (i) to determine, under an objective criterion, if the length of the vector associated with each study plan is the most appropriate for the study we propose and (ii) if the order of inputs may be different. The second approach is the most important in this analysis since it implies that the categories can be analyzed in more than one way but with similar or equivalent results to those that the “pure" content analysis already yielded.
In addition to reducing the previous analysis by contracting the original vector, this also reformulates the established categories, thereby, giving a second aligned analysis guide with greater affinity for the analyzed sample in response. In addition to the contraction of the vectors, this analysis could project the set of vectors into a smaller number of categories that are more robust and consistent than those with which the study began. The interest in conducting this analysis is to establish a method that helps us to objectively filter the content analysis to obtain results that are likely to be interpreted more quickly without losing the consistency of the previous results. To do this, we submitted the data to the algorithm known as k-means, which constitutes an objective and robust analysis of our original data.
Below, we make a simple and brief description of this method. The algorithm’s goal is to cluster data, given an input consisting of a sample of
N data, to generate k subgroups of this sample (clusters) as output. To achieve this,
k centroids or means are initially randomly proposed, then the distance of each of the data to each of these
k centroids is determined, and they are grouped with the centroid closest to the data in question. Generally, an exploration is conducted on the number of clusters—that is, the number of clusters is increased, and the one that minimizes the WCSS (within-cluster sums of squares) parameter, or
inertia, is chosen, seeking that the number of clusters is not too high. A more detailed description of this method can be found in [
77].
Our data was subjected to the
k-means method implemented in the
scikit learn library [
78] within the python programming language, and then a
principal component analysis (PCA was performed and implemented in
scikit learn [
79]), using only two components (to be able to visualize these groups in a diagram). Analyzing the WCSS parameter, we determined that the best compromise between the least variation and the number of desired clusters was 3 as seen in
Figure 1a. It should be noted that, each time the algorithm is executed, the choice of centroids can change. This can be controlled by setting the value of the seed; in our case, we set different values for the seed, and, in
Figure 1b, some of the results obtained are visualized.
Our ensemble consists of a sample of
data, each consisting of a vector formed by the values 0 or 1 (corresponding to the total of the 15 questions, which were initially grouped in one of the content knowledge categories showed in
Table 4). The objective was to know if there is a way to reduce the number of subgroups and whether this is consistent with the current grouping—that is, if any of the new groups completely absorbs any of the categories that were already discussed, or if, on the contrary, it takes elements (questions) from both categories.
Despite the random character present in the algorithm, after performing several simulations with different seed values, we can conclude that, in effect, it is possible to reduce the number of categories (clusters) to 3 because, despite what the image may suggest, detailed analysis allowed us to conclude that, if between one seed and another, the apparent structure of the groups changes, then, on average, the data grouped in cluster 1 in another simulation migrate to group 2 or 3 (similarly for the other two subgroups), and this is only conclusive evidence of the inherent closeness of the data.
Regarding the vector length approach, this methodology tells us that 3 of the 15 questions in the analysis guide can be discarded because they are not significant in the study population, which leads to new vectors of 12 entries. Regarding the second approach, namely a new possible ordering of the entries of the vectors to compose new categories (clusters), we find that these can be reduced to three categories. One of the results that this method yields and confirms the results discussed in section above is that the NCKP-P1, PKNCP-P1, and RP-P1 traits are not integrated in any way in the sample of lesson plans analyzed. For this reason, in
Table 7, the questions corresponding to these traits do not appear in any of the clusters shown. However,
Table 7 shows the questions that comprise the second analysis guide grouped into the new categories that the described method yielded. The gray scale is in correspondence with the colors of the clusters shown in
Figure 1b.
As can be seen, the questions that were not included in any of the clusters found by the algorithm correspond to constructs that the educator forms throughout their practice as a teacher—namely, the meaning of teaching mathematics (RP-P2) and its learning process in preschool (PKNCP-P1) once there is a defined knowledge structure around the concept of numbers (NCKP-P1). These three traits are developed on time scales depending on the work experience in the classroom, adding to a solid conceptual architecture of the numerical knowledge it teaches. Given that the population of educators analyzed is still in the process of teacher training, their experience with the group is expected to be low; and we can say that the abstraction processes necessary in the reflection process have not matured enough.
However, in Group 1, pedagogical and curricular aspects can be observed that organize the educator’s work session around the concept of numbers, according to their class plan: the key learning goals around the development of number sense (NCKP-P2, NCKP-P3, and NCKP-P4), the pedagogy that is known to respond effectively to children’s logic (NCKP-P5, PKNCP-P4, and PKNCP-P5), and elements of curricular understanding, such as the approach and the graduation profile defined for children in the formative field of mathematical thinking (CKNCP-P1, CKNCP-P2, and PKNCP-P2). All these aspects are those that are typically taught during the teacher training stage.
Under the competency development approach in the training plan, the educator constantly practices the knowledge or educational theory that is taught. Therefore, this second cluster (Group 1) has a sense of basic training for teaching and learning. Group 2 refers to the use of the curriculum by requiring the educator to consider the children’s prior knowledge and to align the objective and purposes of the intervention around what is stated in the curriculum used (PKNCP-P3 and CKNCP- P3). Finally, Group 3 resulted in a single-trait cluster (RP-P1), which refers to the permanent process of reflection on educational practice as a modifying agent of constructs that the subject who teaches forms and transforms throughout their practice.
Likely, this new organization of the analysis guide allows us to see theoretical links that were previously were not obvious. For example, Group 1 consists of the pedagogical, instructional, and curricular aspects in which the in-service teacher has been prepared; Group 2 links the curricular organization of the session with the curricular presuppositions of the preschool training path in its entirety and even before it, and finally Group 3 exclusively contains reflective practice as a motor of change and construction of meanings by the teacher in the classroom for continuous improvement.
In this new order, the analysis guide allows us to judge the curriculum under which an educator is trained in Mexico to teach at the preschool level. A substantial load of instructional pedagogical elements is detected in the analyzed population together with general structural elements of the preschool curriculum that it uses (operational aspects) but with a weak understanding of the approaches and vision framed in the basic education curriculum added to a fragile mathematical construction of the number concept. This second analysis is identical to our first discussion of results, which tells us that the implemented method can not only reduce the content analysis technique but also uncover theoretical links that may not have been seen once a new order is detected.
In other words, it shows another face of the same object, making it easier to have a more robust interpretation of the analyzed phenomenon. The method can be used in the same way as shown here with discussion guides leading to vectors of 1s and 0s either for problems similar to the one discussed here or for essentially different ones. Therefore, the PCA used here is valid in populations similar to the one analyzed regardless of the curriculum under which the educator is trained, which constitutes one of the most relevant results of this study.
Finally, for the selected study sample, the questions that make up the only features integrated into the practice of the teachers analyzed are the questions NCKP-P2, NCKP-P5, PKNCP-P4, CKNCP-P1, and RP-P1; grouped into three categories. This reduction of the analysis vector indicates the integrated elements and the categories in which these elements are projected. It is evident that the teacher’s school training, or rather, the curricular response of the study plan in which preschool educators are trained, is responding to minimal needs in the face of the educational reality they face.
Although this vector of 5 entries describes this sample of educators in pre-service, there are 13 entries corresponding to the second analysis guide, and this requires modifications, either in the implementation or in the redesign of the curriculum in normal education, to obtain graduation profiles that, instructionally, are capable of conducting the teaching (in the case at hand) of the concept of numbers in preschool education (the stage of 3–5 years). The proper analysis of Mexican Normal Education curricula is far from the scope of this work; however, it is aligned with our research perspectives.