Artículos

Explorando o conhecimento do conteúdo pedagógico

M-PCK de professores pré-serviço

Exploring mathematical Pedagogical content knowledge of pre-service teachers

Explorando el conocimiento del contenido pedagógico matemático de los futuros maestros

Edgar J. Sintema

(Weizmann Institute of Science, Israel)edgar.sintema@weizmann.ac.il

Mogege David Mosimege

(University of the Free State, South Africa)mosimegemd@ufs.ac.za

Asvi Heris

(DMI-St Eugene University, Zambia)herisasvi@gmail.com

Recibido: 06/07/2023

Aprobado: 06/07/2023

RESUMO

O objetivo deste estudo transversal foi examinar as percepções dos licenciandos sobre seu conhecimento de conteúdo pedagógico matemático e determinar o efeito de variáveis demográficas (gênero, grau de escolaridade) em seu conhecimento de conteúdo pedagógico matemático. Um questionário de escala Likert foi usado para coletar dados de 104 licenciandos. A estatística descritiva e o teste U de Mann-Whitney foram usados para examinar o conhecimento percebido dos licenciandos sobre estratégias de ensino, linguagem e os símbolos matemáticos, sobre o erro (equívocos), o currículo e seu conhecimento percebido dos alunos. Os resultados mostram que, em geral, a maioria dos licenciandos tem maior percepção do seu conhecimento do conteúdo pedagógico matemático. Eles consideram o planejamento de aulas, o propósito do currículo de matemática e as ferramentas de avaliação usadas em uma sala de aula de matemática como aspectos importantes de seu conhecimento percebido do currículo. Também foi identificado que seu conhecimento percebido de equívocos os capacitaria a conhecer e antecipar os equívocos de seus alunos. No entanto, os resultados também indicaram que não houve diferença estatisticamente significativa no conhecimento do conteúdo pedagógico matemático dos licenciandos com base no gênero e no grau de escolaridade. Implicações do estudo também são discutidas.

Palavras-chave: licenciandos. conhecimento pedagógico do conteúdo. matemática.

ABSTRACT

The purpose of this cross-sectional study was to examine pre-service teachers’ perceptions of their mathematical pedagogical content knowledge and to determine effect of demographic variables (Gender, year of study) on their mathematical pedagogical content knowledge. A Likert scale questionnaire was used to collect data from 104 pre-service teachers. Descriptive statistics and Mann-Whitney U-test were used to examine pre-service teachers’ perceived knowledge of teaching strategies, mathematical language and symbols, misconceptions, curriculum, and their perceived knowledge of learners. Results show that in general, majority pre-service teachers have strong perceptions of their mathematical pedagogical content knowledge. They consider lesson planning, purpose of the mathematics curriculum and assessment tools used in a mathematics classroom as important aspects of their perceived knowledge of the curriculum. It was also revealed that their perceived knowledge of misconception would enable them to know and anticipate their learners’ misconceptions. However, results also indicated that there was no statistically significant difference in pre-service teachers’ mathematical pedagogical content knowledge based on gender and year of study. Study implications are also discussed.

Keywords: pre-service teachers. pedagogical content knowledge. mathematics.

RESUMEM

El propósito de este estudio transversal fue examinar las percepciones de futuros docentes sobre su conocimiento del contenido pedagógico matemático y determinar el efecto de las variables demográficas (género, grado de escolarización) en su conocimiento del contenido pedagógico matemático. Se utilizó un cuestionario de escala Likert para recopilar datos de 104 futuros profesores. Se utilizaron estadísticas descriptivas y la prueba U de Mann-Whitney para examinar el conocimiento percibido de los maestros en formación sobre estrategias de enseñanza, lenguaje y símbolos matemáticos, e também para examinar conceptos erróneos, el currículo y la decepciones los professores sobre el conocimiento percibido de los estudiantes. Los resultados muestran que, en general, la mayoría de los futuros docentes tienen una buena percepción con relación a su conocimiento del contenido pedagógico matemático. Consideran la planificación de las clases, el propósito del currículo de matemáticas y las herramientas de evaluación utilizadas en un salón de clases de matemáticas como aspectos importantes de su conocimiento sobre el currículo. También se reveló que su conocimiento percibido de conceptos erróneos les permitiría conocer y anticipar los conceptos erróneos de sus alumnos. Sin embargo, los resultados también indicaron que no hubo una diferencia estadísticamente significativa en el conocimiento del contenido pedagógico matemático de los futuros maestros en función del género y grado de escolaridad. También se discuten las implicaciones del estudio.

Palabras clave: futuros docentes. conocimiento del contenido pedagógico. matemáticas.

Introduction

Teachers’ pedagogical content knowledge (PCK) is important if teaching and learning has to improve (Kadarisma et al., 2019; Sintema & Marban, 2020). For mathematics teachers, mathematical pedagogical content knowledge (M-PCK), which refers to subject (

To develop PCK as conceptualized by Shulman (1986) and to emphasise the concept of M-PCK, Ball et al. (2008) developed the mathematical knowledge for teaching in which they discussed and distinguished teachers’ knowledge of

It is thus important to understand teachers’ own awareness of their M-PCK. This because their perception of students’ mathematical difficulties would determine the extent to which they can help their students overcome these difficulties (Gasteiger et al., 2020; Rahman et al., 2022). Similarly, Teachers’ perception of teaching approaches and strategies would determine their ability to choose strategies that can help then effectively deliver mathematical concepts to their students. Thus, the purpose of this study is to explore pre-service teachers’ (PSTs) M-PCK and whether their M-PCK significantly differs based on their gender and year of study. Several studies have been conducted on teachers’ PCK of mathematics but not much has been investigated on the M-PCK perceptions of PSTs considering that these are the future classroom teachers. This study will contribute to the literature on PSTs’ self-reported perceived M-PCK.

Pedagogical content knowledge for mathematics teachers

Scholars have argued that PCK is useful at two important stages in the teaching and learning process (Alonso & Kim, 2015; Baki & Arslan, 2022). For example, Baki and Arslan, (2022) describe M-PCK in terms of

PSTs’ potential to effectively teach mathematical concepts in their classrooms is highly dependent on their PCK (Aydin-Gunbatar et al., 2020; Baki & Arslan, 2022; Rahman et al., 2021a, 2021b). PSTs require sufficient PCK to adequately understand the mathematics curriculum, formulate measurable objectives and design lesson plans that maximise student participation in the planed classroom activities (Attard et al., 2020; Faikhamta et al., 2020; Rahman et al., 2021a; Song, 2019; Vossen et al., 2019). Several other studies have been carried out on PSTs’ M-PCK (Cheryan et al., 2017; Danisman & Tanisli, 2017; Depaepe et al., 2015; Lee et al., 2018; Rahman et al., 2022; Trobst et al., 2019) with mixed results. For example, some researchers (Depaepe et al., 2015; Trobst et al., 2019) found high M-PCK levels among participants while others (Danisman & Tanisli, 2017; Lee et al., 2018) reported low M-PCK in the participants of their studies. In addition, other studies (Cheryan et al., 2017; Rahman et al., 2022) reported that demographic variables like gender did not influence PSTs’ M-PCK. Considering the importance of demographic variables in understanding teachers’ perceptions, we decided to investigate the effect of gender and year of study on Zambian PSTs. This would contribute contextual perspectives to the literature vis-a-vis the Zambian context. Thus, our study sought answers to the following questions:

1. What are secondary pre-service mathematics teachers’ M-PCK perceptions?

2. To what extent do secondary pre-service mathematics teachers’ M-PCK perceptions differ when analysed based on demographic variables (gender and year of study)?

Methodology

To adequately explore and have a wide understanding of PSTs’ M-PCK perceptions, the current study employed the survey research design (Gable, 1994; Nardi, 2018). Participants of the study were drawn from 3^{rd} and 4^{th} year students pursuing a 4-year bachelor’s degree at two public universities in Zambia using the cluster sampling technique (c.f Sharma, 2017). Both university follow a similar model of mathematics teacher education. In this model, the Department(s) of Mathematics Education only offers pedagogical courses to its PSTs. PSTs register for mathematics content courses in the Department(s) of Mathematics. The universities have no control over gender balance among its students, as every student is free to enrol in a study program of their choice. Cluster sampling has the advantage of selecting subjects that fragmented and occur naturally in their settings, in our case, universities (Acharya et al., 2013; Sharma, 2017). The sample comprised of more male students which speaks to the high enrolment of males in the Bachelor of Science with education (BSc.Ed) program. Considering that participation was voluntary, more 3^{rd} year students (55.8%) volunteered to participate in this study. The sample was dominated by male participants (75%) and half of participants were aged 24 - 26 years old. All these demographic characteristics of participants are presented in Table 1.

Characteristic | N | % | |

Age | 27 years and older | 16 | 15.4 |

24-26 years old | 52 | 50 | |

20-23 years old | 36 | 34.6 | |

Gender | Male | 78 | 75 |

Female | 26 | 25 | |

Year of study | 3^{rd} year |
58 | 55.8 |

4^{th} year |
46 | 44.2 |

Data were collected using a 17-item self-administered M-PCK perceptions questionnaire adapted from Bukova-Guzel et al. (2013). The items were spread across five constructs: knowledge of teaching strategies (KTS) with 3 items, knowledge of mathematical language and symbols (KMLS) with 2 items, knowledge of misconceptions (KM) with 3 items, knowledge of learners (KL) with 2 items, and knowledge of curriculum (KC) with 7 items. We calculated the Cronbach’s alpha reliability coefficient using our sample and found it to be 0.75 which is not good enough but acceptable (George & Mallery, 2003). Participants were requested to indicate their M-PCK perceptions by responding to the questionnaire using a 5-point Likert scale (1 = Never, 2 = rarely, 3 = Undecided, 4 = Usually, 5 = Always).

Data were analysed using the Statistical Package for Social Sciences (SPSS) Version 23. Three analyses were conducted in SPSS. First, we run descriptive statistics (i.e., frequencies, percentages, means, standard deviations) to establish PSTs’ M-PCK perceptions (Research question 1). We then sought to examine the effect of gender and year of study on PSTs’ M-PCK perceptions (Research question 2). We accomplished this by conducting a nonparametric statistical test: Mann-Whitney U-test.

Results: Pre-service mathematics teachers’ M-PCK perceptions

PSTs’ M-PCK perceptions were first analysed using descriptive statistics (see Table 2). Results of this analysis shows that PSTs possess high perceived M-PCK (see Table 2). This implies that they have high confidence in their teaching strategies as well as their ability to identify and solve problems related to learner’s misconceptions. Additionally, these PSTs are highly confident that they would use correct mathematics languages and symbols in their future classrooms. Further, these PSTs are confident about their use of curriculum materials and understanding the demands of the mathematics curriculum. Table 2 shows mean scores and standard deviations the five M-PCK scales that speak to aspects of teaching strategies , mathematical language and symbols , Misconceptions , knowledge of learners and curriculum knowledge .

Table . Descriptive statistics of pre-service teachers’ M-PCK perceptions

Knowledge domain | Mean | Std. Deviation |

Knowledge of Teaching Strategies | 4.20 | .88 |

Knowledge of Mathematical Language and Symbols | 4.49 | .67 |

Knowledge of Misconceptions | 4.16 | .71 |

Knowledge of Learners | 4.07 | .90 |

Knowledge of Curriculum | 4.25 | .53 |

Pre-service mathematics teachers’ M-PCK perceptions based on their KTS

PSTs’ perceptions of their KTS were defined by 3 items (see table 3). It is clear from Table 3 that majority of PSTs that participated in this study perceived mathematics as being closely related to the day-to-day lives of humans. It is the perception of majority PSTs (52.9%) that they can always relate concepts in mathematics to day-to-day real life situation during classroom instruction. Additionally, 33.7% perceived themselves as teachers who would usually relate mathematical concepts to real life situations. Surprisingly, 5.8% of the PSTs indicated that they can never use analogies when teaching mathematics while 2.9% said that they can never design appropriate activities to teach mathematics in their classrooms. Quite a large number (25%) of PSTs were not sure about their use of analogies in explaining mathematical concepts in their classrooms.

Never | Rarely | Undecided | Usually | Always | |||||||||||

F | % | F | % | F | % | F | % | F | % | ||||||

I can design appropriate activities to present mathematical concepts | 3 | 2.9 | 11 | 10.6 | 1 | 1 | 40 | 38.5 | 49 | 47.1 | |||||

I can relate mathematical concepts to daily life in instruction | 0 | 0 | 12 | 11.5 | 2 | 1.9 | 35 | 33.7 | 55 | 52.9 | |||||

I can use analogies to mathematical concepts in instruction | 2 | 5.8 | 31 | 29.8 | 25 | 24 | 29 | 27.9 | 13 | 12.5 |

Pre-service mathematics teachers’ M-PCK perceptions based on their KMLS

According to responses displayed in Table 4, PSTs generally showed high perceptions of their KMLS. This was justified by the large number of PSTs (58.7%) who said that they would always and 33.7% who indicated that they would use proper and acceptable mathematical language when presenting mathematical concepts to their learners. Only 2.9% were not sure and 4.8% would rarely use appropriate mathematical language during instruction. On the other hand, most of the PSTs (60.6%) showed high perceptions of using appropriate and acceptable mathematical symbols during instruction. An additional 33.7% of PSTs perceived themselves teachers who would usually teach using appropriate mathematical symbols. However, a small number of PSTs (2.9%) reported that they were undecided on whether to use mathematical symbols or not while 4.8% said that they would rarely use mathematical symbols in class.

Never | Rarely | Undecided | Usually | Always | |||||||||||

F | % | F | % | F | % | F | % | F | % | ||||||

I can use mathematical language properly when presenting mathematical concepts | 0 | 0 | 5 | 4.8 | 3 | 2.9 | 35 | 33.7 | 61 | 58.7 | |||||

I can use mathematical symbols properly | 0 | 0 | 4 | 3.8 | 2 | 1.9 | 35 | 33.7 | 63 | 60.6 |

Pre-service mathematics teachers’ M-PCK perceptions based on their KM

PSTs’ perceptions of their knowledge of learners’ misconceptions were defined by 3 items (Table 5). In relation to their ability to anticipate students’ difficulties about a given mathematics topic, 88.5% of PSTs indicated that they can either always anticipate (45.2%) or usually anticipate (43.3%) possible students’ difficulties about a given mathematics topic. It is rather surprising that some PSTs (2.9%) would never anticipate their students’ difficulties in a mathematics topic. Interestingly, 43.3% of PSTs said that they would always design or select classroom activities that would not lead their students to develop misconceptions about a given mathematics topic. Similarly, 35.6% of PSTs reported that they were knowledgeable about their students’ possible misconceptions about a given mathematics topic. Only 8.7% said they would rarely do that.

Never | Rarely | Undecided | Usually | Always | |||||||||||

F | % | F | % | F | % | F | % | F | % | ||||||

I can anticipate students’ possible difficulties about a topic | 3 | 2.9 | 2 | 1.9 | 7 | 6.7 | 45 | 43.3 | 47 | 45.2 | |||||

I know students’ possible misconceptions about a topic | 1 | 1 | 9 | 8.7 | 6 | 5.8 | 51 | 49 | 37 | 35.6 | |||||

I can design activities that will not cause students to develop misconceptions about the topic | 1 | 1 | 9 | 8.7 | 10 | 9.6 | 39 | 37.5 | 45 | 43.3 |

Pre-service mathematics teachers’ M-PCK perceptions based on their KL

Table 6 shows PSTs’ responses to items related to their perceptions of KL. Analysing Table 6, majority PSTs (47.1%) perceived themselves as usually being knowledgeable about their students’ prior knowledge about a given mathematics topic while 37.5% said they would always know their students’ prior knowledge of a mathematics topic. On the other hand, more than half of the PSTs (54.8%) indicated that they can choose appropriate examples for students’ developmental levels in my lessons. This generally implies that PSTs had high perceptions of their KL.

Never | Rarely | Undecided | Usually | Always | |||||||||||

F | % | F | % | F | % | F | % | F | % | ||||||

I know students’ prior knowledge about a topic | 2 | 1.9 | 8 | 7.7 | 6 | 5.8 | 49 | 47.1 | 39 | 37.5 | |||||

I can choose appropriate examples for students’ developmental levels in my lessons | 0 | 0 | 6 | 5.8 | 2 | 1.9 | 39 | 37.5 | 57 | 54.8 |

Pre-service mathematics teachers’ M-PCK perceptions based on their KC

PSTs’ perceptions of their KC were defined by 7 items, the most of all compared to other knowledge constructs which were defined by not more than 3 items each. Examining Table 7, a general picture that emerges shows that pre-service teacher had high KC perceptions. For example, the item related to PSTs’ knowledge of designing a lesson plan was the most outstanding with 72.1% of PSTs affirming they knowledge of lesson plan preparation. Similarly, the item related to PSTs’ knowledge of linking goals of a given mathematics topic when designing a lesson plan was yet another outstanding one with 56.7% of PSTs indicating that they would always linking lesson plan design to goals of a mathematics topic. However, the item that related to PSTs’ ability to use assessment tools from the mathematics curriculum raised concern as 18.3% of PSTs reported that they can rarely use assessment tools from a mathematics curriculum.

Never | Rarely | Undecided | Usually | Always | |||||||||||

F | % | F | % | F | % | F | % | F | % | ||||||

I have knowledge about the purposes of the mathematics curriculum | 1 | 1 | 10 | 9.6 | 5 | 4.8 | 38 | 36.5 | 50 | 48.1 | |||||

I can design a lesson plan for a topic | 1 | 1 | 3 | 2.9 | 4 | 3.8 | 21 | 20.2 | 75 | 72.1 | |||||

I plan my lessons so as to relate the purposes of the mathematics curriculum with students’ needs | 1 | 1 | 8 | 7.7 | 4 | 3.8 | 42 | 40.4 | 49 | 47.1 | |||||

When designing my lesson plans, I consider the goals of the topic | 1 | 1 | 4 | 3.8 | 7 | 6.7 | 33 | 31.7 | 59 | 56.7 | |||||

I can use the assessment tools presented in the mathematics curriculum | 1 | 1 | 19 | 18.3 | 3 | 2,9 | 41 | 39.4 | 40 | 38.5 | |||||

I can evaluate the effectiveness of the activities I use in the class for students’ conceptual understanding | 1 | 1 | 9 | 8.7 | 4 | 3.8 | 49 | 47.1 | 41 | 39.4 | |||||

I can draw on the results of my evaluations in designing and adjusting the instruction | 1 | 1 | 7 | 6.7 | 10 | 9.6 | 43 | 41.3 | 43 | 41.3 |

Pre-service mathematics teachers’ M-PCK perceptions based on gender and year of study

To determine PSTs’ differences in M-PCK perceptions based on their gender, we conducted the Mann-Whitney U-test (see Table 8). Results of the U-test showed that there were no significant difference between male and female PSTs’ M-PCK perceptions (U[N_{male} = 78, N_{female} = 26] = 928, z = -.65, p = .516). This could further be confirmed by their median scores which are very close to each other for males (Mdn_{male} = 4.33) and females (Mdn_{female} = 4.22).

Gender | N | Mean rank | Sum of ranks |

Male | 78 | 53.61 | 4181.50 |

Female | 26 | 49.17 | 1278.50 |

To determine PSTs’ differences M-PCK perceptions based on their year of study, we conducted the Mann-Whitney U-test (see Table 9). Results of the U-test showed that there were no significant difference between third and fourth year PSTs’ M-PCK perceptions (U[N_{third year} = 58, N_{fourth year} = 46] = 1262, z = -.47, p = .637). This could further be confirmed by their median scores which are very close to each other for males (Mdn_{third year} = 4.27) and females (Mdn_{fourth year} = 4.31).

Year of study | N | Mean rank | Sum of ranks |

3^{rd} year |
58 | 51.26 | 2973 |

4^{th} year |
46 | 54.07 | 2487 |

Discussion and conclusion

Generally, the M-PCK perceptions of the prospective teachers are strong. This implies that most of the PSTs in our study will have high confidence in the teaching competencies and will be aware of their students’ mathematical needs and difficulties. This has implications for teacher training in Zambia. It means that teacher education programs for mathematics teachers support PSTs to develop their self-efficacy. This result is consistent with findings of Trobst et al. (2019) who found that participants in their study exhibited high levels of PCK. However, findings of this study are not consistent with those of Depaepe et al. (2015), whose study revealed gaps in teacher PCK and content knowledge. PSTs in our study were more confident of their perceptions of KMLS and KC perceptions than they were in other M-PCK sub-factors. This is a sign that they will be confident to handle and use the mathematics curriculum confidently and will also be able to teach using appropriate mathematical language. Results also indicate that PSTs are likely to be confident when selecting teaching strategies. In addition, they are also confidently deal with challenges in identifying their learners’ misconceptions. PSTs of this study are confident that they will clearly understand demands of the mathematics curriculum.

The findings of this study are also not consistent with results of the study by Danisman and Tanisli (2017) whose study was aimed at determining high school mathematics teachers’ PCK of probability. They investigated teachers’ knowledge of content, knowledge of teaching methods and strategies, curriculum knowledge, and knowledge of their students. Results showed that secondary school mathematics teachers had insufficient PCK. Lee et al. (2018) also found low levels of PCK in their study. It was also found that PSTs’ M-PCK did not statistically differ based on demographic variables (gender and year of study), similar to findings of Rahman et al. (2022). This implies that gender has no influence on PSTs’ confidence in their M-PCK. Thus, male and female PSTs would equally succeed in developing their PCK for teaching mathematics with confidence (c.f. Attard et al., 2020; Cheryan et al., 2017; Rahman et al., 2022). This also implies that teacher educators are training teachers that are equally confident in the M-PCK.

As regards yeah of study, it is surprising that PSTs’ M-PCK did not differ significantly. It was expected that PSTs in 4^{th} year would exhibit higher M-PCK than those in 3^{rd} year. The lack of difference does not imply that the teacher education program has no impact on PSTs. Rather it could be because most pedagogical courses are covered in the third year in these universities. It also implies that PSTs in 3^{rd} and 4^{th} year equally feel confident to teach mathematics. Teaching practice is mainly done in 3^{rd} year and all participants had already experienced teaching in secondary school by the time of data collection. Thus, the result could have been influenced by the fact that all participants had already completed similar pedagogical courses and a teaching practice experience. This result is consistent with findings of Rahman et al. (2022) who found that teaching experience did not have an effect on teachers’ M-PCK.

Limitations and recommendations for further studies

Lastly, this study had two major limitations that future studies should overcome. First, the questionnaire that was used contained 17 items to capture M-PCK perceptions where some M-PCK sub-factors like KMLS were composed of only two items. The small number of items per sub-factor were not sufficient to widely capture M-PCK perceptions of PSTs. Future users of the instrument we used should consider adding more items to comprehensively capture M-PCK perceptions. Second, our study was purely quantitative. Future researchers should consider mixed methodology to have in-depth understanding of PSTs’ M-PCK perceptions by incorporating interviews or other qualitative techniques of collecting data. We also hope that future studies will consider having a larger sample than the one we used not only for validation of our study but for easy generalization of results.

Acknowledgement

We thank the ETDP SETA Research Chair in Mathematics Education at the University of the Free State, South Africa, for financial, technical and logistical support throughout the development and compilation. The ETDP SETA Research Chair made the collaboration between authors in writing this manuscript successful.

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