Mathematical knowledge for teaching in Pre-Service teachers from the Complementary Training Program of the Escuela Normal Superior Cristo Rey from Barrancabermeja.

doi:10.21203/rs.3.rs-3232881/v1

This study analyzes the state of mathematical competencies of pre-service teachers of the Complementary Training Program (CTP) of the Escuela Normal Superior Cristo Rey, from Barrancabermeja, Colombia and compares it with the results obtained in a private and an official university. The research has a quantitative, non-experimental, longitudinal approach, having as variable the mathematical knowledge of the students of the sample, from a comparative analysis of the levels of Mathematical Knowledge for Teaching (MKT) at the beginning -first semester- and at the end -fourth semester- of the educational process. The test measures three mathematical competencies: reasoning, communication and problem solving. The results show that in all samples there is low performance in the communicative competence and that students in the first semester of the public university show better competencies than those in the Complementary Training Program (CTP) and private university, in that order. However, at the end of the course, it is the CTP students who show better performance than their public and private university counterparts. This could be due to the emphasis on science and mathematics didactics, as well as the constant exposure to class planning and professional practices. In conclusion, students, future teachers, both in their first and last semesters, need to reinforce their communicative competence in mathematics and their exposure to professional practices should be privileged throughout their careers.

Applied Mathematics

Mathematical knowledge for teaching

Normal Superior School

Pre-service teachers

Mathematics

In what type of institution will undergraduate students receive the best mathematics education? In a public university, in a private university or in a college? How is the learning of mathematics and its didactics compared and evidenced in undergraduate students in different contexts? One of the challenges of universities is to offer quality education to their students, which is not only achieved by having doctors to teach the different subjects, but it is also necessary to give students the tools to develop not only disciplinary knowledge but also pedagogical-didactic knowledge that will lead to a successful professional practice (Ponte, 2012).

Shulman (1986) divided teacher knowledge into three categories: subject matter content knowledge, pedagogical content knowledge, and curricular knowledge. This research seeks to know and compare the levels of three educational institutions in terms of curricular and subject content knowledge, comparing them at the beginning and end of the career and among them, particularly paying attention to a context in which there have not been many previous studies, which is that of the CTP in Colombian Teacher Training Colleges (called in Spanish “Escuelas Normales Superiores”).

Recent prior experiences (Aké, 2021; Barichello & Firer, 2021; Franco-Seguí & Alsina, 2022; Goedert-Doná & Jacques-Ribeiro, 2022; Gualdrón Pinto, Ávila Zárate, & Ordóñez Vargas, 2021; Segarra & Juliá, 2021) have mainly found that the curricular knowledge of mathematics is maintained throughout the undergraduate programs, with slight setbacks or advances, and that what is really strengthened after completing an undergraduate program is the pedagogical and didactic skills of the student body.

In Colombia, teachers are trained at distinct levels and in different institutions. There is training in teacher training schools (Escuelas Normales Superiores), which are official organizations that prepare students for a period of two years to graduate as education professionals. (Asociación Nacional de Escuelas Normales Superiores, 2015). There is also undergraduate training in private and public universities. Until the appearance of this study, there is no recent comparative of the level of mathematical knowledge in teacher training programs in these contexts. This study evaluated the MKT of teachers in the process of training in a public university program, a private university program and the CTP of the Escuela Normal Superior Cristo Rey. It is important for universities to know the level of disciplinary management of their graduates from different degree programs so that successful experiences can be replicated, and improvement plans can be established where necessary.

This is a descriptive study because it seeks to describe the MKT situation of ENSCR students and compare it with the public and private institutions in the previous study. It is also quantitative, since it uses as its main source of information and analysis the numerical results of the mathematical knowledge test for the end of basic secondary school, administered by the Colombian Institute for the Promotion of Higher Education (ICFES). It is also of quasi- experimental design and purposive sampling because the sample was conveniently chosen as the students who were in their first and fourth semesters at the time of the research.

Mathematical knowledge for teaching

MKT is a topic that has been widely studied since the 1980s in the 20th century (e.g., Gualdrón, Ávila, & Ordoñez, 2021; Sosa & Carrillo, 2012; Gómez, 2014). It seeks to understand what are the conceptual tools that a mathematics teacher must have to satisfactorily do his or her job. First, there is the mathematical content knowledge that implies a deep mastery of the mathematical concepts, procedures and structures taught in the school curriculum (Velásquez, Cisneros, & Castro, 2015). That is, fully understanding the contents and the connections between them. In addition, it is crucial to have pedagogical knowledge of the content, which allows teachers to transform mathematical knowledge into accessible and meaningful forms for students (Velásquez & Cisneros, 2013). This involves knowing how to present, explain, and organize mathematical concepts, as well as diagnose and address students' learning difficulties (Hill, et al., 2008).

Another important aspect is knowledge of the curriculum, which implies that teachers know the objectives, standards, and sequences of the mathematics curriculum, as well as knowledge of the educational context: student characteristics, school policies and practices, available resources, and environmental challenges and opportunities (Hill, Ball and Schilling, 2008).

Finally, knowledge of students' learning and developmental processes is critical.

Teachers must have a solid understanding of how students learn mathematics and how they develop cognitively and emotionally. This involves knowing the common stages and obstacles in learning mathematics, as well as using effective strategies to support student development (Shulman, 1986).

Modern didactics of mathematics

At the same time, teachers are expected to approach mathematics teaching from a constructivist approach, in which students build their own knowledge from the interaction with numbers and measurements in their environment, in the context of meaningful mathematical activities, such as problem solving through deep mathematical reasoning. It is also sought that these teachers do not teach mathematics in isolation, but that it be applied to problem solving, understanding that this would promote critical thinking, creativity, and the ability to transfer acquired knowledge to new and/or everyday situations (Skovsmose, 1994; Lesh & Doerr, 2003).

It is also common for ICT to serve as a means for mathematical modeling, with tools such as symbolic calculus software, graphing calculators, and dynamic geometry software.

Additionally, collaborative learning environments are favored, in which all students feel capable towards mathematics according to their roles in problem solving. (Lesh & Doerr, 2003).

Additionally, recent studies have shown that mathematics teachers tend to privilege the cognitive over the attitudinal and procedural skills (Lopez & Alsina, 2016) and that the evaluation style they develop "focuses on the development of merely mechanical procedures" in which the evaluation of affective aspects is "notably less considered" (Cardenas Lizarazo, Blanco Nieto, Guerrero Barona, and Caballero Carrasco, 2016).

Curriculum plan for the subject "Mathematical Thinking" at the Escuela Normal Superior Cristo Rey

The “Escuela Normal Superior Cristo Rey” in Barrancabermeja offers a training space focused on the development of mathematical thinking, within the framework of scientific and technological competence. (Escuela Normal Superior Cristo Rey, 2021). The objective of the program is to train teachers from a humanizing creative approach, capable of addressing educational problems and participating in the construction of a just and democratic society. The objective of this training space is to provide teachers in training with elements to guide the teaching of mathematics, promoting progress and continuity in the learning of students. It seeks to enhance students' scientific and technological thinking skills to solve problematic situations specific to their context.

The training space addresses the importance of mathematics as part of culture and its relevance for students to acquire the knowledge and skills necessary to function in society. Emphasis is placed on the development of mathematical competencies that allow students to interpret and express information clearly, apply mathematical knowledge in everyday situations, use reasoning processes to solve problems and be confident in the use of mathematical tools. The focus of the training space is to provide tools to teachers in training to facilitate the learning of mathematics in preschool and elementary school children, through pedagogical strategies, analysis of curricular guidelines and class observation.

The justification for this training space lies in the need for teachers in training to acquire a solid knowledge of mathematics and develop new teaching strategies to overcome fear and generate genuine interest in students. Furthermore, it is emphasized that the mastery of mathematics is fundamental to understand the complexity of reality and to open doors in society. The theoretical references that support this training space include the problem-solving model of George Polya (1990), the principles and standards for teaching and learning mathematics by Font, Godino, Goñi, & Planas (2011), and the didactics of mathematics in early childhood education by Arteaga Martínez & Macías Sánchez (2016).

The programmatic content of the training space is organized into thematic units, addressing topics such as conceptions about mathematics, teaching-learning models, problem solving and the relationship of mathematics with society. The methodology employed includes the presentation of the training space, the recognition of the students' mathematical knowledge, the analysis of different ways of teaching mathematics and problem solving. The evaluation is based on evidence of knowledge, procedure, and attitude, assessing the mastery of concepts, the development of assigned workshops and student participation.

The educational institution

The present study was conducted in the city of Barrancabermeja, Colombia; at the Escuela Normal Superior Cristo Rey, a state educational institution which offers preschool, elementary, and secondary school, middle school, and the CTP, where this study was applied.

The CTP is offered to students who finish their studies at the institution to take it in a period of four semesters and obtain the title of "Normalista Superior" (Teacher for pre-school and primary), which, under Colombian law, qualifies them as education professional teachers at the preschool and elementary school level. High school graduates from other schools may also take the program, but they must first complete an "introductory" semester, which means that, in their case, the program will last five semesters.

The Escuela Normal Superior Cristo Rey is the most outstanding official educational institution at the municipal level and one of the best at regional level. The purpose of the Escuela Normal Superior Cristo Rey is "to train teachers to perform at the Pre-school and Primary Basic Cycle levels of education through a humanizing pedagogy, which leads to an active commitment to the transformation of the environment." (Escuela Normal Superior Cristo Rey, 2018).

The pedagogical model of the Cristo Rey Teacher Training College is shown in Fig. 1.

This research addresses the research question: What is the MKT of the teachers in training process of the CTP of the Escuela Normal Superior Cristo Rey de Barrancabermeja, Colombia in the year 2022? A quantitative, non-experimental, longitudinal approach was used, since numerical data are collected to test a hypothesis and there is no independent variable; instead, only the context in which the phenomenon develops is observed and analyzed (Hernández-Sampieri, Fernández-Collado, & Baptista-Lucio, 2014).

This design is chosen considering the numerical characteristics of the variable of mastery of disciplinary mathematical knowledge of the students of the CTP, based on a comparative analysis of the MKT at the beginning and at the end of the educational process.

POPULATION AND PARTICIPANTS

The participants chosen for this study are thirty-six first semester students (1 male and thirty-five females) and twenty-six fourth semester students (all females) of the CTP of the Escuela Normal Superior Cristo Rey. These participants were chosen based on the criterion of being in the respective semester at the institution and accepted their voluntary participation in the study. The data obtained in this study were compared with those of the research conducted by Gualdrón, Ávila & Ordóñez (2021) in which the same data collection instrument was used.

DATA COLLECTION TECHNIQUE AND INSTRUMENT

For this research, the survey technique was implemented with a questionnaire instrument that corresponds to some questions in random order, taken from the test booklet with examples of questions of the saber 9° test of mathematics, provided by the ICFES in the year 2015 (ICFES, 2015). This material is generated by this autonomous entity of the Colombian State for the evaluation of education, which guarantees traceability with educational public policies and the validity and reliability of the instrument.

The ICFES evaluates three fundamental competencies in the area of mathematics: Reasoning, Communication and Problem Solving; in turn, each of these competencies is subdivided into three components: spatial-metric, numerical-variational and random, thus categorizing nine diverse types of questions to be evaluated. The questions used in these tests are multiple choice, with four answer options, of which only one is correct. In general, the average time to answer each question is 2 minutes.

In order to privilege the information that could be obtained about the mathematical knowledge and processes used by the students to solve the questionnaire, a space was added to each question to justify the reason for the answer chosen, thus lengthening the response time for each question to an average of 3.7 minutes for each question, with a total time of one hour and forty minutes to complete the 27-question test. In this time, the nine distinct types of competencies and components being assessed were covered. Table 1 shows the distribution and order of the questions in the questionnaire according to the combination of competencies and components.

Table 1

*Distribution of questions by competency and component.*
COMPETITION	COMPONENT	QUESTION NUMBER
REASONING	Numerical- Variational	1, 2, 3
	Random	4, 5, 6
	Spatial metric	7, 8, 9
COMMUNICATION	Spatial metric	10, 11, 12
	Numerical- variational	13, 14, 15
	Random	16, 17, 18
PROBLEM SOLVING	Spatial metric	19, 20, 21
	Numerical- variational	22, 23, 24
	Random	25, 26, 27

VALIDATION OF THE INSTRUMENT

The validity of the instrument is given within the framework of the methodology applied in the development of standardized tests by the ICFES, that is, the evidence-centered design (ECD) model in which the assessment "must construct all important knowledge in the domain of interest and an understanding of how that knowledge is acquired and the use to which it is put" (Mislevy, Almond & Lukas, 2004). These authors highlight the validity offered by the DCE design, in which the line of reasoning is related to the knowledge, skills and abilities that the tests are intended to measure.

The results obtained by the two groups to which the test was applied and the comparison with the results of a previous study are presented below. (Gualdrón Pinto, Ávila Zárate, & Ordóñez Vargas, 2021). In both cases, instructions were given on the application of the instrument in person and, immediately afterwards, the students took the test in its virtual version, through the Google Forms tool. In both groups the same amount of time was given for the resolution of the questions and their justification. In all cases, the justification was mandatory, so that students could support their knowledge or not of the topic they were asked about.

Student scores were compared using the same criteria used by ICFES in 2019; a categorization of skills according to the score obtained, as shown in Table 2:

Table 2

*Performance levels.*
Performance level	Score	Description
Insufficient	0 to 35	The student at this level can probably read punctual information (a piece of data, for example) related to everyday situations and presented in tables or graphs with explicit scale, grid or, at least, horizontal lines: but may have difficulties when comparing different sets of data, involving different variables, or analyzing situations far from their daily life.
Minimum	36 to 50	In addition to what is described in the previous level, the student at this level is able to compare and establish relationships between the data presented, and to identify and extract local and global information in a direct way.

Satisfactory

51 to 70

In addition to what is described in the previous levels, the student at this level selects information, points out errors and performs several types of transformations and simple arithmetic and

algebraic manipulations.

Advanced

71 to 100

In addition to what is described in the previous levels, the student at this level solves problems and justifies the truth or falsity of

statements that require the use of concepts.

Note. Based on ICFES (2017).

MKT FOR THE FIRST HALF OF THE YEAR

A total of thirty-six first semester students took the test; of these students, two were from private schools and the rest from public schools. No student took the test before the end of the time. These results are shown in Table 3.

Table 3

*Results of the first semester students of the Complementary Training Program.*
First semester Complementary Training Program
COMPETENCIES
REASONING			COMMUNICATION			PROBLEM SOLVING
60,49%			50,93%			56,48%
COMPONENT
Numeric-Variational	Random	Spatial metric	Spatial - metric	Numeric-Variational	Random	Spatial metric	Numeric-Variational	Random
57,41%	61,11%	62,96%	60,19%	46,30%	46,30%	55,56%	53,70%	60,19%

Total number of answers correctly justified in

the test

Unjustified or incorrectly justified answers

Justified not knowing how to solve the question

38,99%

61,01%

4,12%

It becomes evident that the reasoning and problem solving competencies reach satisfactory levels, which is explained due to the previous contact that students have had with mathematical exercises throughout their previous school life, but, on the other hand, it is seen that it reaches the upper limit value of the minimum performance in that of communication, given that students have not been forced in many scenarios for the communication of the conceptualization, process or resolution of mathematical problems to peers or students in previous processes. As will be seen below, the mathematics communication item should improve as the teacher-in-training is exposed to more pedagogical practice and contact with students.

It is also shown that seven of the nine components reach satisfactory levels and two (those of the weakest competency, communication) had minimum performance levels.

Regarding justification, more than half of the questions were incorrectly justified or not justified and only 38.99% of the questions were correctly justified. In this sense, it is reaffirmed that students in this semester have greater problems when communicating and justifying mathematical procedures and results.

MKT OF THE FOURTH SEMESTER

On the other hand, twenty-six fourth semester students applied the test. Two of them with training in private educational institutions and twenty-four of them with previous context in public educational institutions. These students are already in the last level of the program and have seen all the academic areas, including "development of mathematical thinking" and "development of scientific thinking". Some of them have even done their research project with a topic related to the didactics of mathematics.

The time used for testing proved to be sufficient for this group. Students in this semester show better performances than their first semester peers in all three competencies and outperform their peers by up to twenty-one percentage points in the reasoning component. The same happens with the components, in which each of the measures of this semester outperform their first semester peers. This information is summarized in Table 4.

Table 4

*Results of the fourth semester students of the Complementary Training Program.*
Fourth semester Complementary Training Program
COMPETENCIES
REASONING			COMMUNICATION			PROBLEM SOLVING
82,05%			64,96%			74,79%
COMPONENT
Numeric-Variational	Random	Spatial - metric	Spatial - metric	Numeric-Variational	Random	Spatial - metric	Numeric-Variational	Random
85,90%	71,79%	88,46%	80,77%	50,00%	64,10%	71,79%	73,08%	79,49%
Total number of answers correctly justified in the test			Unjustified or incorrectly justified answers			Justified not knowing how to solve the question
51,44%			20,78%			2,37%

With respect to justification, there is a higher percentage of justified responses and a drastic decrease in the number of unjustified responses, with a decrease of forty-one percentage points, which could indicate greater conceptual mastery and more developed competencies in fourth semester students.

COMPARISON OF THE MKT OF THE COMPLEMENTARY TRAINING PROGRAM WITH UNDERGRADUATE PROGRAMS AT PUBLIC AND PRIVATE UNIVERSITIES

First semester CBP students have better performance in the three competencies than private university students and lower performance than first semester public university students presented in the study by Gualdrón Pinto, Ávila Zárate, & Ordóñez Vargas (2021). This is evidenced in Fig. 2.

It can be argued that the CBP students come from a long trajectory in the institution, most of the students who continue in the CBP, did their high school and middle school in the school and it is characterized by its good academic level as demonstrated in the national exam, where they obtain superior performance (ICFES, 2022). On the other hand, it is known that the private university accepts students from diverse schooling profiles and previous contexts, as long as they meet the entrance requirements, and it is also known that the public university has a much more demanding admission process, which is probably why those who enter these programs are above the academic performance of the national average.

Regarding the last semester, it is observed that the fourth semester has a much higher performance in the three competencies, compared to the last semesters of both the public and private universities. Competencies such as reasoning (82.05%) far surpass the public university, which in this item registers 68.72%, and the private university, which for the same item reports 53.44%. The superior performance of the CTP students is also repeated in the other two competencies. It is possible to affirm that the students who are about to graduate see very concrete contents, with much application in the classroom and that their process of acquiring didactic and conceptual tools of mathematics is successful.

Figure 3 shows a close comparison of the competencies of the first and fourth semesters of complementary training.

It is thus observed that, during the CTP, students acquire knowledge not only of mathematical didactics, but also some disciplinary knowledge, especially from module 1 of the training space "Development of Mathematical Thinking", in what has to do with everyday mathematical problems. With the exception of the communication component (which changes from minimal to satisfactory), a student who is finishing the fourth semester of complementary training presents an advanced level of performance in reasoning and problem-solving skills.

Communication and problem-solving competencies have similar levels between students who complete the CTP and those who complete the bachelor's degree program at the public university. That is, professionals in both settings will have similar performances in their professional life with respect to communication and problem solving, but a CTP student will have developed more of the reasoning competency.

STATISTICAL ANALYSIS WITH CHI-SQUARE

A tendency to repetition of results of some components was observed with respect to the competency analyzed; for this reason, the degree of dependence or relationship between the ways of proceeding (competencies) and the conceptual and structural aspects (components) of mathematics in the students who were the object of this study was verified in this research. To this end, the chi-square analysis technique was employed, which is used to determine whether there is an association between two categorical variables; that is, whether there is a relationship between them. This test is used to contrast the null hypothesis, which states that there is no statistically significant difference in the distribution of the data between the two variables.

The following is a brief description of the procedure conducted: first, the null hypothesis was established, which was to establish the independence of the two variables to be analyzed (competencies and components) by comparing the results obtained from the statistic with the reference value found in the chi-square distribution table. Then, the validity of this independence was determined or not: if the calculated result was higher than the reference value of the table, the null hypothesis was not validated and the dependence or relationship between the variables was determined; if the result was lower, the null hypothesis was validated and the independence between the variables was established. In the second step, the table of what was observed was elaborated: in it, the frequencies obtained from the three competencies versus the three components were crossed, giving a three-by-three table, which made it possible to calculate the expected values for each component in each of the competencies. The third step was to calculate all the values that, when added together, yielded the calculated chi- square. Each of these values was obtained by dividing the square of the difference between each observed value and each expected value by each expected value, as shown in the following equation:

$${X}^{2}=\sum _{i=1}^{k}\frac{{({O}_{i}-{E}_{i})}^{2}}{{E}_{i}}$$

To obtain the value of the chi-square distribution percentile table, two things were needed: first, to define the percentage of error, which, for this type of study, is 5%; and second, to calculate the degrees of freedom. This value was obtained by multiplying the number of columns minus one by the number of rows minus one, which in our case is equal to: (3 − 1) ∗ (3 − 1) = 4. With these two values, the chi-square value (9.488) was sought in the distribution table, which was compared with the calculated chi of the two groups in order to validate, or not, the null hypothesis.

Table 5

*Chi-square calculated for the data obtained.*
Chi-square distribution table reference value	Calculated Chi-square
Chi-square distribution table reference value	First semester CTP	Fourth semester CTP
9,488	2,69	4,72

According to Table 5, in both cases the calculated chi is lower than the reference value of the distribution table; therefore, the null hypothesis is validated: "There is independence between the components and competencies variables". This means that there is no direct relationship between the competencies and the evaluated components.

It was possible to characterize the MKT in teachers in training process of the CTP of the Escuela Normal Superior Cristo Rey de Barrancabermeja, Colombia in the year 2022. It was found that first semester students start the program with satisfactory levels of reasoning and problem solving, but with deficiencies in communication competence, i.e., first semester students are not sufficiently familiar with mathematical language, concepts, and mathematical tools to explain and understand the problems. This also indicates the difficulty they present when interpreting graphs, tables, and diagrams to understand and communicate information effectively.

As for the fourth semester students, it is shown that the knowledge acquired in the course on didactics of mathematics and the mental maturity of the students leads to an evolution in the three competencies and a significant advance of twenty-one percentage points in the reasoning component. However, it is still noticeable that the most lagging competency of the three is communication. This is worrisome, considering that it is the teachers who should have full mastery of the mathematical communicative competence to guide the students' learning process.

First semester Normal School students show better scores in the mathematics test than their private university counterparts, but a lower performance than public university students, but, on the other hand, already in the fourth semester, they show a significantly better performance than the other two groups of students. It can be inferred that the CTP approach in the didactics of science and mathematics and the need to prepare classes and perform in professional practices during the four semesters benefits mathematical competencies in the face of professional practice as elementary school teachers.

It is recommended that the institutions under study reinforce the communicative competence of mathematics at all levels of training within the career, since it was evidenced that this competence repeatedly obtains the lowest scores in the six groups studied. Also, it is indicated to the institutions to privilege the pedagogical practice as a means to manifest the competences of the didactics of mathematics, understanding that, in the analyzed groups, the teaching of mathematics in the field builds more competences than only the study of the theory of mathematical didactics.

Acknowledgements There are no acknowledgements for this paper. This paper has been drafted by the authors named above.

Author contribution The authors named in this submission have an equal share in contributing to drafting this manuscript.

Funding Open Access funding enabled and organized by the University of the Ballearic Islands (Universitat de les Illes Balears)

Ethical approval Prior to the commencement of the participant selection process, the Chief Investigator sought and gained Institutional Ethics Committee approval. All students of the sample were 18 or older, the legal age of majority in Colombia.

Informed consent A project information Google form was sent to students in the sample at participating schools to outline the scope and procedure of the project. Using an opt-out process, students gave their consent to participate by completing an anonymous, online survey.

Competing interests The authors declare no competing interests.

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Mathematical knowledge for teaching in Pre-Service teachers from the Complementary Training Program of the Escuela Normal Superior Cristo Rey from Barrancabermeja.

Status:

Version 1

Abstract

Figures

INTRODUCTION

THEORETICAL FRAMEWORK

Mathematical knowledge for teaching

Modern didactics of mathematics

Curriculum plan for the subject "Mathematical Thinking" at the Escuela Normal Superior Cristo Rey

The educational institution

METHODOLOGY

POPULATION AND PARTICIPANTS

DATA COLLECTION TECHNIQUE AND INSTRUMENT

VALIDATION OF THE INSTRUMENT

RESULTS AND ANALYSIS

MKT FOR THE FIRST HALF OF THE YEAR

MKT OF THE FOURTH SEMESTER

STATISTICAL ANALYSIS WITH CHI-SQUARE

CONCLUSIONS

Declarations

References

Status:

Version 1